PRINCIPLES  OF 
ELECTRICAL  DESIGN 


McGraw-Hill  BookCompany 


Electrical  World         The  Engineering  and  Mining  Journal 
Engineering  Record  Engineering  News 

Railway  Age  Gazette  American  Machinist 

Signal  E,ngin<?9r  American  Engineer 

Electric  Railway  Journal  Coal  Age 

Metallurgical  and  Chemical  Engineering  P  o  we r 


PRINCIPLES 

OF 

ELECTRICAL  DESIGN 

D.  C.  AND  A.  C.  GENERATORS 


BY 
ALFRED  STILL 

PROFESSOR  OF  ELECTRICAL  DESIGN,  PURDUE  UNIVERSITY;    FELLOW  AMERICAN  INSTITUTE 
OF  ELECTRICAL  ENGINEERS;    MEMBER  INSTITUTION  OF  CIVIL  ENGINEERS;    MEMBER 
INSTITUTION  OF  ELECTMCAL  ENGINEERS.      AUTHOR  OF  "OVERHEAD  ELEC- 
TRIC  POWER  TRANSMISSION,"  "POLYPHASE   CURRENTS,"  ETC. 


Fr<is7 


McGRAW-HILL  BOOK  COMPANY,  INC. 
239  WEST  39TH  STREET.     NEW  YORK 


LONDON:  HILL  PUBLISHING  CO.,  LTD. 

6  A  8  BOUVERIE  ST.,  E.  C. 

1916 


58 


COPYRIGHT,  1916,  BY  THE 
McGRAW-HiLL  BOOK  COMPANY,  INC. 


THB.  MAPLE.  PRESS.  YORK.  PA 


PREFACE 

This  book  is  intended  mainly  for  the  use  of  students  following 
courses  in  Electrical  Engineering,  and  for  this  reason  emphasis  is 
laid  on  fundamentals  and  principles  of  general  application,  while 
but  little  attention  is  paid  to  the  needs  of  the  practical  designer, 
who  may  be  trusted  to  devise  his  own  time-saving  methods  of 
calculation,  provided  always  that  he  has  a  thorough  understand- 
ing of  the  essentials  governing  all  electrical  design. 

The  writer  is  a  firm  believer  in  the  advantage  of  having  a 
concrete  mental  conception  of  the  hidden  actions  which  produce 
visible  or  measurable  results,  and  in  studying  the  electromotive 
forces  developed  in  the  windings  of  electric  generators,  he  con- 
sistently represents  the  effects  as  being  due  to  the  cutting  by  the 
conductors  of  imaginary  magnetic  lines. 

No  attempt  has  been  made  to  deal  adequately  with  the  me- 
chanical principles  involved  in  the  design  of  electrical  machinery. 
Thus  as  a  reference  book  for  the  designer,  this  text  is  admittedly 
incomplete.  It  is  incomplete  also  as  a  means  of  giving  the  stu- 
dent what  he  is  supposed  to  get  from  a  course  in  electrical  design, 
for  the  simple  reason  that  no  art  can  be  mastered  by  the  mere 
reading  of  a  book.  In  this  as  in  every  other  study,  all  that  is 
worth  having  the  student  must  himself  acquire  by  giving  his 
mind  to  the  business  on  hand  and  taking  pains.  The  book  cannot 
do  more  than  serve  as  a  reference  text  or  the  basis  for  a  course 
of  lectures;  and  for  every  hour  of  book  study,  four  to  six  hours 
should  be  spent  in  the  actual  working  out  of  practical  designs. 

The  writer  has  ventured  to  express  some  views  regarding  the 
qualifications  of  the  successful  designer  in  an  introductory  chap- 
ter where  he  believes  they  are  less  likely  to  remain  permanently 
buried  than  if  embodied  in  a  preface  of  unconventional  length. 
At  the  same  time  he  does  not  claim  that  the  procedure  here 
adopted  is  such  as  will  meet  the  requirements  of  the  professional 
designer;  but  in  criticising  the  book,  it  is  important  to  bear  in 
mind  that  its  main  object  is  to  illustrate  the  logical  application 
of  known  fundamental  principles,  and  so  help  the  reader  to 
realize  the  practical  value  of  theoretical  knowledge.  It  is  not  to 

338041 


vi  PREFACE 

be  supposed  for  a  moment  that  an  experienced  designer  can  afford 
the  time  required  to  work  through  the  detailed  design  sheets  as 
here  given  in  connection  with  the  numerical  examples ;  but,  apart 
from  the  fact  that  he  generally  makes  use  of  existing  patterns 
and  stampings,  in  connection  with  which  he  has  at  hand  a  vast 
amount  of  accumulated  data,  he  is  in  a  position  to  apply  short- 
cut methods  to  his  work.  This  is  not  readily  done  by  the  student, 
who  usually  lacks  the  experience,  judgment,  and  sense  of  propor- 
tion, without  which  "rule  of  thumb"  methods  and  rough  ap- 
proximations cannot  be  applied  intelligently. 

Portions  of  the  material  here  presented  have  appeared  recently 
in  articles  and  papers  contributed  by  the  writer  to  the  "Elec- 
trical World,"  the  Journal  of  the  Franklin  Institute,  and  the 
Journal  of  the  Institution  of  Electrical  Engineers;  but  what  has 
been  borrowed  from  these  publications  has  to  a  large  extent  been 
rewritten. 

The  thanks  of  the  writer  are  also  due  to  Mr.  D.  L.  Curtner, 
not  only  for  assistance  in  reading  and  correcting  proofs;  but 
also  for  valuable  suggestions  and  helpful  criticism. 

LAFAYETTE,  IND., 
June,  1916. 


CONTENTS 

PREFACE    v 

LIST  OF  SYMBOLS xi 

CHAPTER  I 

INTRODUCTORY 1 

CHAPTER  II 

THE  MAGNETIC  CIRCUIT — ELECTROMAGNETS 
ART.  PAGE 

1.  The  magnetic  circuit 10 

2.  Definitions 12 

3.  Effect  of  iron  in  the  magnetic  circuit 15 

4.  Magnetic  circuits  in  parallel 18 

5.  Calculation  of  leakage  paths 22 

6.  Flux  leakage  in  similar  designs 27 

7.  Leakage  coefficient 28 

8.  Tractive  force 29 

9.  Materials — Wire  and  insulation 32 

10.  Calculation  of  magnet  windings 41 

11.  Heat  dissipation — Temperature  rise 44 

12.  Intermittent  heating 46 

CHAPTER  III 
THE  DESIGN  OF  ELECTROMAGNETS 

13.  Introductory 48 

14.  Short  stroke  tractive  magnet 49 

15.  Magnetic  clutch 50 

16.  Horseshoe  lifting  magnet  (Numerical  example) 53 

17.  Circular  lifting  magnet  (Numerical  example) 64 

CONTINUOUS  CURRENT  GENERATORS 
CHAPTER  IV 

DYNAMO   DESIGN — FUNDAMENTAL  CONSIDERATIONS — BRIEF  OUTLINE  OF 

PROBLEM 

18.  Generation  of  e.m.f 70 

19.  The  output  formuU 72 

20.  Number  of  poles — Pole  pitch — Frequency 78 

vii 


viii  CONTENTS 

CHAPTER  V 

ARMATURE  WINDINGS  AND  SLOT  INSULATION 
ART.  PAGE 

21.  Introductory 83 

22.  Ring-  and  drum-wound  armatures 84 

23.  Multiple  and  series  windings 84 

24.  Equalizing  connections  for  multiple-wound  armatures 90 

25.  Insulation  of  armature  windings 91 

26.  Number  of  teeth  on  armature 93 

27.  Number  of  commutator  segments — Potential  difference  between 

segments 93 

28.  Nature  and  thickness  of  slot  insulation 94 

29.  Current  density  in  armature  conductors 96 

30.  Length  and  resistance  of  armature  winding 97 

CHAPTER  VI 

LOSSES  IN  ARMATURES — -VENTILATION — TEMPERATURE  RISE 

31.  Hysteresis  and  eddy-current  losses  in  armature  stampings  ...  100 

32.  Usual  densities  and  losses  in  armature  cores 104 

33.  Ventilation  of  armatures 105 

34.  Cooling  surfaces  and  temperature  rise  of  armature 107 

35.  Summary,  and  syllabus  of  following  chapters 113 

CHAPTER  VII 
FLUX  DISTRIBUTION  OVER  ARMATURE  SURFACE 

36.  Air-gap  flux  distribution  with  toothed  armatures 115 

37.  Actual  tooth  density  in  terms  of  air-gap  density 119 

38.  Correction  for  taper  of  tooth 121 

39.  Variation  of  permeance  over  pole  pitch — Permeance  curve    .    .    .123 

40.  Open-circuit  flux  distribution  and  m.m.f.  curves 128 

41.  Practical  method  of  predetermining  flux  distribution 129 

42.  Open-circuit  flux  distribution  curves  as  influenced  by  tooth  satura- 

tion   132 

43.  Effect  of  armature  current  in  modifying  flux  distribution  .    .    .    .135 

CHAPTER  VIII 
COMMUTATION 

44.  Introductory . 140 

45.  Theory  of  commutation 142 

46.  Effect  of  slot  flux    .    .    .    ......    .....    . '" '.    ...  ,.:..-•    .    •  149 

47.  Effect  of  end  flux    .    .    .    J  , '.'..    ,  .Jh 151 

48.  Calculation  of  end  flux 155 

49.  Calculation  of  slot  flux  cut  by  coil  during  commutation 160 

50.  Commutating  interpoles 165 

51.  Example  of  interpole  design 171 


CONTENTS  ix 

ART.  PAGE 

52.  Prevention  of  sparking — Practical  considerations 175 

53.  Mechanical  details  affecting  commutation 178 

54.  Heating  of  commutator — Temperature  rise 181 

CHAPTER  IX 

THE  MAGNETIC  CIRCUIT — DESIGN  OF  FIELD  MAGNETS — EFFICIENCY 

55.  The  magnetic  circuit  of  the  dynamo 185 

56.  Leakage  factor  in  multipolar  dynamos 186 

57.  Calculation  of  total  ampere-turns  required  on  field  magnets   ...  187 

58.  Arrangement  and  calculation  of  field  windings 190 

59.  Temperature  rise  of  field  coils 193 

60.  Efficiency 195 

CHAPTER  X 
DESIGN  OF  A  CONTINUOUS  CURRENT  DYNAMO — NUMERICAL  EXAMPLE 

61.  Introductory 199 

62.  Design  sheets  for  75-kw.  multipolar  dynamo 200 

63.  Numerical  example — Calculations 204 

64.  Design  of  continuous  current  motors 235 

ALTERNATING-CURRENT  GENERATORS 

CHAPTER  XI 
DESIGN  OF  ALTERNATORS — FUNDAMENTAL  CONSIDERATIONS 

65.  Introductory 237 

66.  Classification  of  synchronous  generators 238 

67.  Number  of  phases 239 

68.  Number  of  poles — Frequency 241 

69.  Usual  speeds  of  A.C.  generators 241 

70.  E.m.f.  developed  in  windings 242 

71.  Star  and  mesh  connections 244 

72.  Power  output  of  three-phase  generators 247 

73.  Usual  voltages 248 

74.  Pole  pitch  and  pole  arc 248 

75.  Specific  loading 250 

76.  Flux  density  in  air  gap 251 

77.  Length  of  air  gap — Inherent  regulation 252 

CHAPTER  XII 
ARMATURE  WINDINGS — LOSSES  AND  TEMPERATURE  RISE 

78.  Types  of  windings 255 

79.  Spread  of  winding 257 

80.  Insulation  of  armature  windings 258 

81.  Current  density  in  armature  conductors    .    .    . 259 


x  CONTENTS 

AKT.  PAGE 

§2.  Tooth  and  slot  proportions 259 

83.  Length  and  resistance  of  armature  winding 260 

84.  Ventilation 261 

85.  Full  load  developed  voltage 261 

86.  Inductance  of  A.C.  armature  windings 263 

87.  Calculation  of  armature  inductance 264 

88.  Total  losses  to  be  radiated  from  armature  core 266 

89.  Temperature  rise  of  armature 267 

CHAPTER  XIII 
AIR-GAP  FLUX  DISTRIBUTION — WAVE  SHAPES 

90.  Shape  of  pole  face 269 

91.  Variation  of  permeance  over  pole  pitch  (Salient  pole  machines) .    .271 

92.  M.m.f.  and  flux  distribution  on  open  circuit  (Salient  pole  machines)  272 

93.  Special  case  of  cylindrical  field  magnet  with  distributed  winding .  272 

94.  Armature  m.m.f  in  alternating-current  generators 274 

95.  Slot  leakage  flux 281 

96.  Calculation  of  slot  leakage  flux 284 

97.  Effect  of  slot  leakage  on  full-load  air-gap  flux 286 

98.  Method  of  determining  position  of  armature  m.m.f 288 

99.  Air-gap  flux  distribution  under  load 290 

100.  Form  of  developed  e.m.f.  wave 291 

101.  Form  factor 294 

102.  Equivalent  sine  waves 295 

CHAPTER  XIV 
REGULATION  AND  EFFICIENCY  OF  ALTERNATORS 

103.  The  magnetic  circuit 299 

104.  Regulation 301 

105.  Factors  influencing  the  inherent  regulation  of  alternators  ....  303 

106.  Regulation  on  zero  power  factor 305 

107.  Short-circuit  current 308 

108.  Regulation  on  any  power  factor 309 

109.  Influence  of  flux  distribution  on  regulation 311 

110.  Outline  of  procedure  in  calculating  regulation  from  study  of  e.m.f. 

waves 312 

111.  Efficiency 317 

CHAPTER  XV 
EXAMPLE  OF  ALTERNATOR  DESIGN 

112.  Introductory >    . 320 

113.  Single-phase  alternators 321 

114.  Design  sheets  for  8000-kw.  turbo-alternator 322 

115.  Numerical  example — Calculations 324 

INDEX  .   359 


LIST  OF  SYMBOLS 

A  =  area  of  cross-section ;  area  of  surface. 

A  =  area  of  one  lobe  of  periodic  wave  plotted  to  polar  coordinates. 

Ac  =  ampere-conductors  in  pole  pitch. 

a  =  coefficient  in  resistance-temperature  formulas. 

B  =  magnetic  flux  density  (gauss). 
/?„  or  B"  =  magnetic  flux  density  (maxwells  per  square  inch). 

Ba  =  instantaneous  average  value  of  air-gap  density  over  the  arma- 
ture conductors  of  one  phase. 

Bc  =  average  air-gap  density  in  zone  of  commutation. 
Bc  =  average  or  equivalent  flux  density  in  pole  cores. 
Bg  =  flux  density  in  air  gap  (gausses). 
Bg  =  average  air-gap  density  over  tooth  pitch. 
Bp  =  average  air-gap  density  under  commutating  pole. 
Bt  =  actual  flux  density  in  teeth. 
B.  &  S.  =  Brown  and  Sharpe  wire  gage. 
6  =  number  of  brush  sets. 

C  =  electrostatic  capacity  (farads). 
c  —  coefficient  of  friction. 
c  =  length  of  pole  (radial). 

D  =  diameter  of  armature  core  (including  teeth). 
De  =  diameter  of  commutator  (inches). 

d  =  diameter  of  magnet  core. 

d  —  inside  diameter  of  armature  core. 

d  =  maximum  value  of  equivalent  sine-wave. 

d  =  depth  of  armature 'slot  (length  of  tooth). 
d,  =  equivalent  length  of  tooth. 

E  =  electromotive  force  (e.m.f.);  difference  of  potential  (volt). 
Ea  =  volts  per  phase  in  armature  winding. 
EC  =  volts  per  conductor. 
Ee  =  e.m.f.  generated  in  end  connections  of    short-circuited  coil 

during  commutation  (volts). 
Em  =  mean  or  average  value  of  e.m.f. 
Eo  =  terminal  voltage  when  load  is  thrown  off. 
E,  =  reactance  voltage  drop  per  phase  (slot  leakage  only). 
Et  =  terminal  voltage. 
e  =  instantaneous  value  of  e.m.f.  (volts). 
ee  =  instantaneous  e.m.f.  per  armature  inductor, 
e.m.f.  =  electromotive  force. 

/  =  frequency  (cycles  per  second). 

xi 


xii  LIST  OF  SYMBOLS 

H  =  intensity   of   magnetic    field;   magnetizing   force  (gilberts  per 

centimeter;    or  gauss). 
hp.  =  horsepower. 

/  =  current,  amperes. 
70  =»  current  per  phase  (or  per  conductor)  in  alternator  armature 

winding. 

Ic  =  current  per  conductor  in  armature. 
Im  =  mean  or  average  value  of  current. 
i  =  instantaneous  value  of  current. 

is  =  instantaneous  value  of  current  in  armature  conductor  (for  slot 
leakage  calculations). 

k.v.a.  =  kilovolt-amperes. 
k.w.  =  kilowatts. 

k  =  a  constant;  any  whole  number. 

k  =  cooling  coefficient. 

k  =  distribution  factor  (A.C.  armature  windings). 

pole  arc. 
10  armature  core  length. 

permissible  current  density  at  brush  tip. 
average  current  density  over  brush  contact  surface. 

L  =  inductance;  coefficient  of  self-induction  (henry). 
Lc  =  length  of  cylindrical  surface  of  commutator  (inches). 

I  =  a  length,  usually  expressed  in  centimeters. 

I  =  %le  (commutation). 
I'  =  axial  projection  of  armature  end  connections  beyond  slot  (A.C. 

generator). 

I"  =  a  length  expressed  in  inches. 
la  =  gross  length  of  armature  core. 
lc  =  total  axial  length  of  brush  contact  surface. 
le  =  length  of  end  portions  of  armature  coil. 
If.  =  leakage  factor. 

In  =  net  length  of  iron  in  armature  core  (usually  inches). 
IP  =  axial  length  of  commutating  pole  face. 
lv  —  total  axial  length  taken  up  by  vent  ducts. 

M  =  thickness  of  commutator  mica. 
(M )  =  circular  mils  per  ampere. 
(m)  =  circular  mils, 
m.m.f.  =  magnetomotive  force  (gilbert). 

TV  =  number  of  revolutions  per  minute. 
N,  =  number  of  revolutions  per  second. 
n  =  number  of  radial  air  ducts  in  armature  core. 
n  =  number  of  slots  per  pole. 
ns  =  number  of  slots  per  pole  per  phase. 


LIST  OF  SYMBOLS  xiii 

P  =  power;  watts. 
P  =  permeance. 

P  =  pressure,  pounds  per  square  inch. 
p  =  number  of  poles. 
Pi  =  number  of  electrical  paths  in  parallel  in  armature, winding. 

Q  =  quantity  of  electricity. 

q  =  specific  loading  of  armature  periphery  (ampere-conductors  per 
inch). 

R  =  resistance  (ohm). 

R  -  resistance  of  armature  coil  undergoing  commutation. 

R  =  magnetic  reluctance  (oersted). 
R"  =  resistance  between  opposite  faces  of  an  inch  cube  of  copper 

(ohm). 

Re  —  brush  contact  resistance  per  square  inch  of  contact  surface. 
Rd  =  radial  depth  of  armature  stampings  below  slots. 
jRo  =  resistance  at  temperature  zero  degrees. 
Rt  =  resistance  at  temperature  t  degrees. 

pole  arc. 
r  =  ratio  - 

pole  pitch. 

r  =  length  of  radius  vector. 

S  =  number  of  turns  in  a  coil  of  wire. 
iS>  =  brush  contact  surface. 
s  =  slot  width, 
a/.  =  space  factor. 
SI  =  ampere  turns. 

(SI)a  =  armature  ampere-turns  per  pole. 
(SI)g  =  ampere-turns  for  air  gap. 

T  =  temperature  rise  in  degrees  Centigrade. 
T  =  number  of  turns  in  armature  coil. 

Te  =  number  of  turns  in  armature  coil  between  tappings  to  com- 
mutator bars. 
T.  =  number  of  inductors  in  one  slot. 

t  =  interval  of  time. 

t  =  temperature  (degrees  Centigrade). 

t  =  thickness  of  magnet  winding. 

t  =  width  of  tooth. 
te  =  time  of  commutation. 
tr  =  width  of  rotor  tooth. 

» 

V  =  peripheral  velocity  of  armature — centimeters  per  second. 
Ve  =  surface  velocity  of  commutator — centimeters  per  second. 

v  =  peripheral  velocity — feet  per  minute. 

ve  =  peripheral  velocity  of  commutator  surface — feet  per  minute. 
v*  =  average  velocity  of  air  in  ventilating  ducts — feet  per  minute. 


xiv  LIST  OF  SYMBOLS 

W  =  power — watts. 
W  =  width  of  brush  (circumferential). 
Wa  =  brush  width  (arc)  referred  to  armature  periphery. 
w  =  a  portion  of  the  total  (circumferential)  width  of  brush. 


cooling  coefficients  (armature  temperature). 

X  =  reactance  (ohm). 

Z  =  impedance  (ohm). 

Z  =  number  of  inductors  in  series  per  phase  (A.C.  generator). 
Z  =  number  of  inductors  on  armature  (D.C.  generator). 
Z'  =  total  number  of  inductors  (A.C.  generator). 

a  =  angle  denoting  slope  of  coil  side  in  end  connections  of  armature 

winding. 
a  =  angle  of  phase  displacement  of  developed  e.m.f.  due  to  armature 

cross  magnetization. 

/3  =  angle  of  lag  of  current  behind  phase  of  open-circuit  e.m.f. 
A  =  current  density;  amperes  per  square  inch. 
Aw  =  maximum  current  density  over  contact  surface  of  brush. 
5  =  clearance  between  coil  sides  in  end  connections  of  armature 

windings. 

5  =  length  of  actual  air  gap — tooth  top  to  pole  face. 

be  =  length  of  equivalent  air  gap  in  machines  with  toothed  armatures. 
8S  =  deflection  of  shaft  (inches). 

6  =  an  angle. 

6  =  angle  of  lag  (cos  6  =  power  factor). 

X  =  slot  pitch. 

n  =  permeability  =  B/H^ 

•n-  =  3.1416  approximately. 

T  =  pole  pitch  (usually  in  inches). 

$  =  magnetic  flux — (maxwell). 

<l>  =  total  flux  entering  armature  from  each  pole  face. 
$0  =  flux  per  pole  actually  cut  by  armature  conductors. 
$>c  =  total  flux  entering  teeth  comprised  in  commutating  zone. 
&d  =  flux  entering  armature  core  through  roots  of  teeth  (commuta- 
tion). 

$e  =  total  flux  cut  by  one  end  of  armature  coil  during  commutation. 
3>e  =  total  flux  cut  by  end  connections  (both  ends)  of  polyphase 

armature  winding. 
3>s  =  "equivalent"  slot  leakage  flux  (magnetic  circuit  closed  through 

roots  of  teeth). 

>'ts  =  "equivalent"  slot  flux  (magnetic  circuit  closed  through  tops  of 
teeth). 

$i  =  leakage  flux  (maxwells). 

$,  =  total  slot  leakage  flux. 

<l>x  =  flux  entering  armature  in  space  of  one  tooth  pitch. 

^  =  internal  power-factor  angle. 
^  =  "apparent"  internal  power-factor  angle. 

co  =  2*f 


PRINCIPLES 
OF  ELECTRICAL  DESIGN 


CHAPTER  I 
INTRODUCTORY 

By  devoting  a  whole  chapter  to  introductory  remarks  and 
generalities  which  are  rarely  given  a  prominent  place  in  modern 
technical  literature,  the  author  hopes  not  only  to  explain  the 
scheme  and  purpose  of  this  book,  but  to  show  what  may  be 
gained  by  an  intelligent  study  of  the  conditions  to  be  met,  and 
the  difficulties  to  be  overcome,  by  the  designer  of  electrical 
machinery. 

The  knowledge  required  of  the  reader  includes  elementary 
mathematics,  the  use  of  vectors  for  representing  alternating 
quantities,  the  principles  of  electricity  and  magnetism,  and 
some  familiarity  with  electrical  apparatus  and  machinery, 
such  as  may  be  acquired  in  the  laboratories  of  teaching  institu- 
tions equipped  for  the  training  of  electrical  engineers,  or  in  the 
handling  and  operation  of  electrical  plant  in  manufacturing 
works  and  power  stations.  The  principles  of  the  magnetic 
circuit  will  be  explained  here  in  some  detail,  because  the  whole 
subject  of  generator  design  from  the  electrical  standpoint  is 
little  more  than  a  practical  application  of  the  known  laws  of 
the  electric  and  magnetic  circuits;  but  a  fair  knowledge  of  the 
physics  underlying  the  action  of  electromagnetic  apparatus  is 
presupposed. 

The  conception  of  the  magnetic  circuit  consisting  of  closed 
lines  or  tubes  qf  induction  linked  with  the  electric  circuit — 
involving  the  cutting  of  these  magnetic  lines  by  the  conductors 
in  which  an  e.m.f.  is  generated — is  unquestionably  a  useful  one 
for  the  practical  engineer;  and  the  student  should  endeavor  to 
form  a  mental  picture  of  these  imaginary  magnetic  lines  in 
connection  with  every  piece  of  electrical  apparatus  or  machinery 
which  he  desires  to  understand  thoroughly. 

1 


2  PRINCIPLES  OF  ELECTRICAL  DESIGN 

Having  clearly  realized  the  general  shape  and  distribution 
of  the  magnetic  field  surrounding  a  conductor  or  linked  with  a 
coil  of  wire  carrying  an  electric  current,  the  next  step  is  to  cal- 
culate with  sufficient  accuracy  for  practical  purposes  the  quantity 
of  magnetic  flux  produced  by  a  given  current;  or  the  e.m.f. 
developed  by  the  cutting  of  a  known  magnetic  field.  This  leads 
to  the  consideration  of  units  of  measurement. 

The  practical  units  of  the  C.G.S.  system  will  be  used  so  far  as 
possible;  but  since  engineers  of  English-speaking  countries  still 
prefer  the  foot  and  inch  for  the  measurement  of  length,  there 
must  necessarily  be  a  certain  amount  of  conversion  from  centi- 
meter to  inch  units,  and  vice  versa.  This  may,  at  first  sight, 
appear  objectionable;  but,  in  the  opinion  of  the  writer,  there  is 
something  to  be  gained  by  having  to  transform  results  from  one 
system  of  units  to  another.  The  process  helps  to  counteract 
the  tendency  of  mathematically  trained  minds  to  lay  hold  of 
symbols  and  formulas  and  treat  them  as  realities,  instead  of 
striving  always  to  visualize  the  physical  (or  natural)  reality 
which  these  symbols  stand  for.  The  same  may  be  said  of  such 
alphabetical  letters  as  are  in  general  use  to  denote  certain  physical 
quantities  or  coefficients;  as  /*  for  permeability,  and  L  for  the 
coefficient  of  self-induction.  Familiarity  with  these  symbols 
tends  to  obscure  the  physical  meaning  of  the  things  they  stand 
for;  and  although  uniformity  in  the  use  of  symbols  in  technical 
literature  cannot  be  otherwise  than  advantageous,1  the  use  of 
unconventional  symbols  involves  their  correct  definition,  and 
for  this  reason  their  occasional  appearance  in  writings  that  are 
professedly  of  an  instructional  nature  should  not  be  condemned. 
This  point  is  made  here  to  emphasize  the  writer's  'conviction 
that  the  student  should  endeavor  to  regard  symbols  and  mathe- 
matical analysis  as  convenient  means  to  attain  a  desired  end; 
and  that  he  should  cultivate  the  habit  of  forming  a  concept  or 
mental  image  of  the  physical  factors  involved  in  every  problem, 
even  during  the  intermediate  processes  of  a  calculation,  if  this 
can  be  done. 

By  way  of  illustrating  the  application  of  fundamental  magnetic 
principles,  the  design  of  electromagnets  will  be  taken  up  before 
considering  the  magnetic  field  of  dynamos.  This  preliminary 
study  should  be  very  helpful  in  paving  the  way  to  the  main 
subject;  and  the  chapter  on  magnet  design  has  been  written  with 

1  A  list  of  the  symbols  used  will  be  found  at  the  beginning  of  this  book. 


INTRODUCTORY  3 

this  end  in  view :  it  does  not  treat  of  coreless  solenoids  or  magnetic 
mechanisms  with  relatively  long  air  gaps;  because  the  air  clear- 
ance is  always  small  in  dynamo-electric  machinery. 

In  the  method  of  design  as  followed  in  this  book,  an  attempt  is 
made  to  base  all  arguments  on  scientific  facts,  and  build  up  a 
design  in  a  logical  manner  from  known  fundamental  principles. 
This  is  admittedly  different  from  the  method  followed  by  the 
practical  designer,  who  uses  empirical  formulas  and  "  short 
cuts,"  justified  only  by  experience  and  practical  knowledge. 
It  must  not,  however,  be  supposed  that  a  commercial  machine 
can  be  designed  without  the  aid  of  some  rules  and  formulas 
which  have  not  been  developed  from  first  principles,  for  the 
simple  reason  that  the  factors  involved  are  either  so  numerous 
or  so  abstruse  that  they  cannot  all  be  taken  into  account  when 
deriving  the  final  formula  or  equation.  In  any  case  the  con- 
stants used  in  all  formulas,  even  when  developed  on  strictly 
scientific  lines,  are  invariably  the  result  of  observations  made  on 
actual  tests;  and  many  of  them,  such  as  the  coefficients  of  fric- 
tion, magnetic  reluctance,  and  eddy-current  loss,  are  subject  to 
variation  under  conditions  which  it  is  difficult  to  determine. 
The  formulas  used  in  design  are  therefore  frequently  empirical, 
and  they  yield  results  that  are  often  approximations  only;  but 
an  effort  will  be  made  to  explain,  whenever  possible,  the  scientific 
basis  underlying  all  formulas  used  in  this  book. 

A  perception  of  the  fitness  of  a  thing  to  fulfil  a  given  purpose 
and  of  the  relative  importance  of  the  several  factors  entering 
into  a  problem,  is  essential  to  the  successful  designer.  This 
quality,  which  may  be  referred  to  as  engineering  judgment,  is 
not  easily  taught;  it  grows  with  practice,  and  is  strengthened  by 
the  experience  gained  sometimes  through  repeated  failures; 
but  it  is  necessary  to  success  in  engineering  work,  whether  this 
is  of  the  nature  of  invention  and  designing,  or  the  surmounting 
of  such  obstacles  and  difficulties  as  will  arise  in  every  branch  of 
progressive  engineering.  All  the  conditions  and  governing  factors 
are  not  accurately  known  at  the  outset,  and  a  good  designer  is 
able  to  make  a  close  estimate  or  a  shrewd  guess  which,  in  nine 
cases  out  of  ten,  will  give  him  the  required  proportion  or  dimen- 
sion; he  will  then  apply  tests  based  upon  established  scientific 
principles  in  order  to  check  his  estimate,  and  so  satisfy  himself 
that  his  machine  will  conform  with  the  specified  requirements. 


4  PRINCIPLES  OF  ELECTRICAL  DESIGN 

A  knowledge  of  the  theory  and  practice  of  design,  the  thorough- 
ness of  which  must  depend  upon  the  line  of  work  to  be  ultimately 
followed,  would  seem  to  be  of  great  importance  to  every  engineer. 
It  may  not  be  of  great  benefit  to  all  men  in  the  matter  of  forming 
judgment  and  developing  ingenuity  or  inventiveness;  but  it 
will  at  least  help  to  bridge  the  gap  between  the  purely  academic 
and  logically  argued  teachings  of  the  schools,  and  the  methods  of 
the  practical  engineer,  who  requires  results  of  commercial  value 
quickly,  with  sufficient,  but  not  necessarily  great,  accuracy,  and 
who  usually  depends  more  upon  his  intuition  and  his  quickness 
of  perception,  than  upon  any  logical  method  of  reasoning. 

It  must  not  be  thought  that  these  remarks  tend  to  belittle  or 
underrate  the  method  of  obtaining  results  through  a  sequence  of 
logically  proven  steps;  on  the  contrary,  this  is  the  only  safe 
method  by  which  the  accuracy  of  results  can  be  checked,  and  it  is 
the  method  which  is  followed,  whenever  possible,  throughout 
this  book.  It  is  not  by  the  reading  of  any  book  that  the  art  of 
designing  can  be  learned;  but  only  by  applying  the  information 
gathered  from  such  reading  to  the  diligent  working  out  of 
numerical  examples  and  problems. 

Although  the  work  done  in  the  drafting  room  is  not  necessarily 
designing,  it  does  not  follow  that  the  designer  need  know  nothing 
about  engineering  drawing.  The  art  of  making  neat  sketches 
or  clear  and  accurate  drawings  of  the  various  parts  of  a  machine, 
is  learnt  only  by  practice;  yet  every  engineer,  whatever  line  of 
work  he  may  follow,  should  be  able  not  only  to  understand  and 
read  engineering  drawings,  but  to  produce  them  himself  at  need. 
It  is  particularly  important  that  he  should  be  able  to  make  neat 
dimensioned  sketches  of  machine  parts,  because,  in  addition  to 
the  practical  value  of  this  accomplishment,  it  is  an  indication 
that  he  has  a  clear  conception  of  the  actual  or  imagined  thing, 
and  can  make  his  ideas  intelligible  to  others.  Clear  thinking  is 
absolutely  essential  to  the  designer.  He  must  be  able  to  visualize 
ideas  in  his  own  mind  before  he  can  impart  these  ideas  to  others. 
Young  men  seldom  realize  the  importance  of  learning  to  think, 
neither  do  they  know  how  few  of  their  elders  ever  exercise  their 
reasoning  faculty.  The  man  who  can  always  express  himself 
clearly,  either  in  words  or  by  sketches  and  drawings,  is  invariably 
one  whose  thoughts  are  limpid  and  who  can  therefore  realize  a 
clear  mental  picture  of  the  thing  he  describes.  The  ability  to 


INTRODUCTORY  5 

"see  things"  in  the  mind  is  an  attribute  of  every  great  engineer. 
Vagueness  of  thought  and  mental  inefficiency  are  revealed  by 
untidy  and  inaccurate  sketches,  poor  composition  and  illegible 
writing.  It  is,  however,  possible  to  train  the  mind  and  greatly 
increase  its  efficiency  by  developing  neatness  and  accuracy  in 
the  making  of  sketches,  and  by  the  study  of  languages. 

The  knowledge  of  foreign  languages  has  an  obvious  practical 
value  apart  from  its  purely  educational  advantage,  but  the  study 
of  English,  for  the  engineer  of  English-speaking  countries,  is  of 
far  greater  importance.  By  enlarging  and  enriching  one's 
vocabulary  through  the  reading  of  high  class  literature,  and  by 
paying  constant  attention  to  the  correct  meaning  of  words  and 
their  proper  connection  in  spoken  and  written  language,  the 
clearness  of  thought  important  to  every  engineer,  and  essential 
to  the  designer,  may  be  cultivated  to  an  extent  which  the  average 
student  in  the  technical  schools  and  engineering  universities 
entirely  fails  to  recognize.  In  an  address  delivered  on  April  8, 
1904,  to  the  Engineering  Society  of  the  University  of  Nebraska, 
DR.  J.  A.  L.  WADDELL  said, 

"Too  much  stress  cannot  well  be  laid  on  the  importance  of  a  thorough 
study  of  the  English  language.  Given  two  classmate  graduates  of  equal 
ability,  energy,  and  other  attributes  contributary  to  a  successful  career, 
one  of  them  being  in  every  respect  a  master  of  the  English  language 
and  the  other  having  the  average  proficiency  in  it,  the  former  is  certain  to 
outstrip  the  latter  materially  in  the  race  for  professional  advancement." 

Considering  further  the  difference  between  the  training  re- 
ceived by  the  student  in  the  schools  and  the  training  he  will 
subsequently  receive  in  the  world  of  practical  things,  it  must 
be  remembered  that  the  object  of  technical  education  is  mainly 
to  develop  the  mind  as  a  thinking  machine,  and  provide  a  good 
working  basis  of  fundamental  knowledge  which  shall  give  weight 
and  balance  to  all  future  thinking.  The  commercial  aspect  of 
engineering  is  seen  more  clearly  after  leaving  school  because  it 
is  not  easily  taught  in  the  class  room.  The  student  does  not, 
therefore,  get  a  proper  idea  of  the  value  of  time.  Engineering 
is  the  economical  application  of  science  to  material  ends,  and  if 
the  items  of  cost  and  durability  are  omitted  from  a  problem,  the 
results  obtained — however  important  from  other  points  of  view 
—have  no  engineering  value.  The  cost  of  all  finished  work, 
including  that  of  the  raw  materials  used  in  construction,  is  the 


6  PRINCIPLES  OF  ELECTRICAL  DESIGN 

cost  of  labor.  Provided  the  work  is  carefully  done,  the  element 
of  time  becomes,  therefore,  of  the  greatest  importance.  A 
student  in  a  technical  school  may  be  able  to  produce  a  neat  and 
correct  drawing,  but  the  salary  he  could  earn  as  a  draughtsman 
in  an  engineering  business  might  be  very  small  because  his  rate 
of  working  will  be  slow.  The  designer  must  always  have  in 
mind  the  question  of  cost,  not  only  material  cost — which  is 
fairly  easy  to  estimate — but  also  labor  cost,  which  depends  on 
the  size  and  complication  of  parts,  accessibility  of  screws  and 
bolts,  and  similar  factors.  These  things  are  rarely  learned 
thoroughly  except  by  actual  practice  in  engineering  works,  but 
the  student  should  try  to  realize  their  importance,  and  bear 
them  in  mind.  A  study  of  design  will  do  something  toward 
teaching  a  man  the  value  of  his  time.  Thus,  although  it  is 
important  to  check  and  countercheck  all  calculations,  and  time 
so  spent  is  rarely  wasted,  yet  it  is  essential  to  know  what  degree 
of  approximation  is  allowable  in  the  result.  This  is  a  matter  of 
judgment,  or  a  sense  of  the  absolute  and  relative  importance  of 
things,  which  is  developed  only  with  practice.  What  is  worth 
doing,  what  is  expedient,  and  what  would  be  mere  waste  of  time, 
may  be  learned  surely,  if  slowly,  by  the  study  and  practice  of 
machine  design. 

It  is  by  taking  on  responsibilities  that  confidence  and  self- 
reliance  are  developed;  and  the  student  may  work  out  examples 
in  design  by  following  his  own  methods,  regardless  of  the  par- 
ticular practice  advocated  by  a  book  or  teacher.  He  can  usually 
check  his  results  and  satisfy  himself  that  they  are  substantially 
correct.  This  will  give  him  far  more  encouragement  and 
satisfaction  than  the  blind  application  of  proven  rules  and  for- 
mulas. By  making  mistakes — that  are  frequently  due  to 
oversights  or  omissions — and  by  having  to  go  over  the  ground  a 
second  or  third  time  in  order  to  rectify  them,  an  important 
lesson  is  learned,  namely,  that  one  must  resist  the  tendency  to 
jump  at  conclusions.  The  necessity  of  checking  one's  work, 
and  proceeding  systematically  by  doing  at  the  right  time  and  in 
the  right  place  the  particular  thing  that  should  be  done  before 
all  others,  is  of  great  value  in  developing  one  of  the  most  im- 
portant qualifications  of  the  engineer.  This  has  already  vaguely 
be^n  referred  to  as  engineering  judgment,  a  sense  of  proportion, 
seeing  the  fitness  of  things;  but  all  these  are  allied,  if  not  actually 
identical,  with  the  one  faculty  of  inestimable  value  known  as 


'    INTRODUCTORY  7 

common  sense,  so  called — according  to  the  definition  of  a  witty 
Frenchman — because  it  is  the  least  common  of  the  senses. 

It  should  be  realized  clearly  that  the  true  designer  is  a  maker, 
not  an  imitator.  The  function  of  the  designer  is  to  create.  His 
value  as  a  live  factor  in  the  engineering  world  will  increase  by 
just  so  much  as  he  rises  above  the  level  of  the  mere  copyist. 
The  man  who  can  see  what  has  to  be  done,  and  how  it  may  be 
done,  is  always  of  greater  value  than  the  man  who  merely  does  a 
thing,  however  skilfully,  when  the  manner  of  doing  it  has  been 
explained  to  him. 

In  addition  to  a  sound  knowledge  of  engineering  principles 
and  practice,  a  designer  should  preferably  have  a  leaning  toward 
original  investigation  or  research  work.  He  should  not  be  bound 
by  the  trammels  of  convention,  nor  discouraged  by  the  ground- 
less belief  that  what  has  been  done  before  has  necessarily  been  done 
rightly.  On  the  contrary,  he  should  assert  his  personality,  and 
have  the  courage  of  his  own  opinions,  provided  these  are  based, 
and  intelligently  formed,  on  established  fundamental  principles, 
the  truth  and  soundness  of  which  are  undeniable. 

If  the  chief  function  of  the  designing  engineer  is  to  create, 
the  cultivation  of  the  imagination  is  obviously  of  the  utmost 
value.  This  is  a  point  that  is  frequently  overlooked.  In  other 
creative  arts,  such  as  poetry  and  painting,  intuition  and  a  fertile 
imagination  are  considered  essential  to  success,  and  there  can  be 
ho'  valid  reason  for  undervaluing  the  possession  of  these  qualities 
by  the  engineer.  The  work  of  the  designer  is  artistic  rather 
than  purely  scientific;  that  is  to  say  it  requires  skill  and  ingenuity 
in  addition  to  mere  knowledge.  Without  a  sound  basis  of 
engineering  knowledge,  the  designer  is  not  likely  to  succeed, 
because  his  conceptions,  like  those  of  many  so-called  inventors, 
would  have  no  practical  application;  but  it  is  also  true  that  the 
great  designers,  even  of  mechanical  and  electrical  machinery,  do 
not  always  understand  why  they  have  done  a  certain  thing  in  a 
certain  way.  They  work  by  intuition  rather  than  by  methods 
that  are  obviously  logical,  but  their  early  training  and  thorough 
knowledge  of  engineering  facts  and  practice  act  as  a  constant 
and  useful  check,  with  the  result  that  they  rarely  make  mistakes 
of  serious  importance. 

It  is  not  suggested  that  the  exalted  moods  and  "inspired 
imaginings"  of  the  poet  or  artist  would  be  of  material  advantage 


8  PRINCIPLES  OF  ELECTRICAL  DESIGN 

to  the  practical  engineer;  but  the  present  writer  wishes  to  state, 
most  emphatically,  that,  in  his  opinion,  the  average  engineer 
does  not  rate  imagination  at  its  proper  value,  neither  does  he 
cultivate  it  as  he  might,  did  he  realize  the  advantages — if  only 
from  a  grossly  commercial  standpoint — that  would  thereby 
accrue.  There  is  to-day  a  tendency  to  underestimate  the 
value  of  abstract  speculation  and  the  pursuit  of  any  study 
or  enterprise  of  which  the  immediate  practical  end  is  not  obvious. 
The  fact  that  the  indirect  benefit  to  be  derived  therefrom  may, 
and  generally  does,  greatly  outweigh  the  apparent  advantages 
of  so-called  utilitarian  lines  of  study,  is  generally  overlooked. 
It  is  an  admitted  fact  that  the  outlook  of  the  graduate  from 
many  of  the  engineering  schools  is  narrow:  this  is  no  doubt 
largely  due  to  faults  in  the  system  and  the  teachers;  but  the 
student  himself  is  apt  to  neglect  his  opportunities  for  the  study 
of  subjects  such  as  general  literature,  languages,  history,  and 
political  economy,  on  the  plea  that  he  would  be  wasting  his  time. 
It  is  only  at  a  later  period  of  his  engineering  career  that  he 
begins  to  realize  how  an  intelligent  and  appreciative  study  of 
these  broader  subjects  would  have  stimulated  his  mind  and 
cultivated  his  imagination  to  a  degree  which  would  be  a  great 
and  lasting  benefit  to  him  in  his  profession. 

It  is  unfortunate  that  neither  the  nature  of  the  subject  nor  the 
manner  in  which  it  is  presented  in  the  following  chapters  is 
likely  to  stimulate  the  imaginative  faculty;  but  the  writer  be- 
lieves that  no  apology  is  needed  for  referring  in  this  chapter  to 
subjects  outside  the  scope  of  the  main  portion  of  the  book. 
In  presenting  fundamental  principles  and  showing  how  they  may 
be  applied  to  the  design  of  machines,  it  is  necessary  to  arrange 
the  matter  in  accordance  with  some  logical  scheme;  and  it  is 
just  because  a  book  such  as  the  present  one  cannot  give,  and 
does  not  claim  to  give,  all  that  goes  to  the  making  of  a  designing 
engineer,  that  it  was  deemed  advisable  to  say  something  of 
a  general  nature  relating  to  the  art  of  designing  electrical 
machinery. 

The  principles  underlying  the  action  of  dynamo-electric 
machinery  may  be  studied  under  two  main  headings: 

1.  The  magnetic  condition  due  to  an  electric  current  in  a 
conductor  or  exciting  coil. 

2.  The  e.m.f.  developed  in  a  conductor  due  to  changes  in 
the  magnetic  condition  of  the  surrounding  medium. 


INTRODUCTORY  9 

This  last  effect,  which  may  be  attributed  to  the  cutting  of  the 
magnetic  lines  by  the  electric  conductors,  will  be  considered 
when  taking  up  the  design  of  dynamos.  For  the  present  it  will 
be  advisable  to  investigate  condition  (1)  only,  and  Chaps. 
II  and  III  will  be  devoted  to  the  study  of  the  magnetic  circuit; 
to  the  calculation  of  the  excitation  required  to  produce  a  given 
magnetic  flux;  or,  alternatively,  the  quantitative  determination 
of  the  flux  when  the  size,  shape,  and  position,  of  the  exciting 
coils  are  known. 


CHAPTER  II 
THE  MAGNETIC  CIRCUIT— ELECTROMAGNETS 

In  all  dynamo-electric  machinery,  coils  of  wire  carrying  electric 
currents  produce  a  magnetic  field  in  the  surrounding  medium— 
whether  air  or  iron — and  the  purport  of  this  chapter  is  to  show 
how  the  magnetic  condition  due  to  an  electric  current  can  be 
determined  within  a  degree  of  accuracy  generally  sufficient  for 
practical  purposes. 

The  design  of  the  magnetic  circuit  of  dynamo-electric  genera- 
tors does  not  differ  appreciably  from  the  design  of  electromagnets 
for  lifting  or  other  purposes,  and  it  is,  therefore,  proposed  to 
consider,  in  the  first  place,  the  fundamental  principles  and 
calculations  involved  in  proportioning  and  winding  electro- 
magnets to  fulfil  specified  requirements.  Particular  attention 
will  be  paid  to  types  of  magnets  with  small  air  gaps  because  these 
serve  to  illustrate  the  conditions  met  with  in  field-magnet  design, 
and  the  principles  of  the  magnetic  circuit  can  be  applied  with 
but  little  difficulty;  while,  in  the  case  of  coreless  solenoids  or 
magnets  with  very  large  air  gaps,  the  p&ths  of  the  magnetic  flux 
cannot  readily  be  predetermined,  and  empirical  formulas  or 
approximate  methods  of  calculation  have  to  be  used.  When  the 
magnetic  circuit  is  mainly  through  iron,  and  the  air  gaps  are 
comparatively  short,  it  is  generally  possible  to  picture  the  lines 
or  tubes  of  magnetic  flux  linking  with  the  electric  circuit,  thus 
facilitating  the  quantitative  calculation  of  the  flux  at  various 
parts  in  the  circuit;  but  when  the  path  of  the  magnetic  lines 
is  largely  through  air  or  other  "non-magnetic"  material,  the 
analogy  between  the  magnetic  and  electric  circuits  is  less  con- 
venient and  may  indeed  lead  to  confusion;  the  quantitative 
calculations  become  more  difficult  and  less  scientific,  calling  for 
an  experienced  designer  if  results  of  practical  value  are  desired. 

1.  The  Magnetic  Circuit. — Without  dwelling  on  the  mathe- 
matical conceptions  of  the  physicist,  which  may  be  studied  in 
all  books  on  magnetism,  it  may  be  stated  without  hesitation 
that  the  analogy  between  the  magnetic  and  electric  circuits, 
and  the  idea  of  a  closed  magnetic  circuit  linked  with  every  electric 

10 


THE  MAGNETIC  CIRCUIT— ELECTROMAGNETS    11 

circuit,  will  be  most  useful  to  the  designer  of  electrical  machinery. 
The  magnetic  flux  is  thought  of  as  consisting  of  a  large  number  of 
tubes  of  induction,  each  tube  being  closed  upon  itself  and  linked 
with  the  electric  circuit  to  which  the  magnetic  condition  is 
due.  The  distribution  of  the  magnetic  field  will  depend  upon  the 
shape  of  the  exciting  coils,  and  upon  the  quality,  shape,  and 
position,  of  the  iron  in  the  magnetic  circuit.  The  amount  of 
the  magnetic  flux  in  a  given  magnetic  circuit  will  depend  upon 
the  m.m.f.  (magnetomotive  force)  and  therefore  on  the  current 
and  number  of  turns  of  wire  in  the  exciting  coils. 

OHM'S  law  for  the  electric  circuit  can  be  put  in  the  two  forms: 

e.m.f.  T       E 

(a)  Current  =  —  r—     — ,  or  /  =  „ 

resistance  R 

(6)   Current  =  e.m.f.  X  conductance,  or  I  =  E  X  ( B) 

Similarly,  in  the  magnetic  circuit: 

magnetomotive  force 
(a)  Magnetic  flux  of  induction  =  - 

magnetic  reluctance 

or 

*  -  ™^t  (1) 

(6)  Magnetic  flux  =  magnetomotive  force  X  permeance 
or  <*>  =  m.m.f.  X  P  (2) 

In  this  analogy,  4>  is  the  total  flux  of  induction,  usually  ex- 
pressed in  C.G.S.  lines,  or  maxwells;  m.m.f.  is  the  force  tending 
to  produce  the  magnetic  condition — expressed  in  ampere-turns 
(Jjte  engineer's  unit)  or  in  gilberts — the  C.G.S.  unit;  and  reluctance 
is  the  magnetic  equivalent  of  resistance  in  the  electric  circuit. 
It  is  necessary  to  bear  in  mind  that  although  these  are  funda- 
mental formulas  of  the  greatest  value  in  the  calculation  of  mag- 
netic circuits,  yet  they  are  based  on  an  analogy  which,  with 
all  its  advantages,  has  its  limitations.  The  chief  difference 
between  OHM'S  law  of  the  electric  circuit  and  the  analogous 
expression  as  applied  to  the  magnetic  circuit  lies  in  the  fact  that 
the  magnetic  reluctance  does  not  depend  merely  upon  the 
material,  length,  and  cross-section,  of  the  various  parts  of  the 
magnetic  circuit,  but  also — when  iron  is  present — on  the  amount 
of  the  flux,  or,  more  properly,  on  the  flux  density,  which  is  an 
important  factor  in  the  determination  of  the  permeability  (/*). 


12  PRINCIPLES  OF  ELECTRICAL  DESIGN 

2.  Definitions. — Magnetomotive  Force. — The  difference  of  mag- 
netic potential  which  tends  to  set  up  a  flux  of  magnetic  induction 
between  two  points  is  called  the  magnetomotive  force  (m.m.f.) 
between  those  points.  The  unit  m.m.f. — known  as  the  gilbert — 
will  set  up  unit  flux  of  induction  between  the  opposite  faces  of 
a  centimeter  cube  of  air.  If  we  consider  any  closed  tube  of 
induction  linked  with  a  coil  of  S  turns  carrying  a  current  of 
/  amperes,  the  total  ampere  turns  producing  this  induction  are 
SI,  and  the  total  m.m.f.  is, 

4?r 
m.m.f.  =  JQ  SI  gilberts.1 

Magnetizing  Force. — The  magnetomotive  force  per  centimeter 
is  called  the  magnetizing  force,  or  magnetic  force.  The  symbol  H 
is  generally  used  to  denote  this  quantity  which  is  also  referred  to 
as  the  intensity  of  the  magnetic  field,  or  simply  field  intensity, 
at  the  point  considered.  The  magnetomotive  force  is,  therefore, 
the  line  integral  of  the  magnetizing  force,  or, 

m.m.f.  =  2#SZ 

where  dl  is  a  short  portion  of  the  magnetic  circuit — expressed  in 
centimeters — over  which  the  magnetizing  force  H  is  considered  of 
constant  value.  Thus  H  =  QAir  X  ampere-turns  per  centimeter 

SI 
=  0.47T-7-  or,  if  it  is  preferred   to  use   ampere-turns   per  inch 

(not  uncommon  in  engineering  work),  we  may  write  H  =  0.495 
(SI  per  inch) . 

1  What  the  practical  designer  wants  to  know  is  the  number  of  ampere- 

47T 

turns  required  to  produce  a  given  magnetic  flux.     The  factor  y^  constantly 

enters  into  magnetic  calculations  as  it  is  required  to  convert  the  engineer's 
unit  (ampere-turn)  into  the  C.G.S.  unit  (gilbert).  It  should  not  be  neces- 
sary to  explain  its  presence  here,  because  this  is  done  more  or  less  lucidly  in 
most  textbooks  of  physics.  It  should  be  sufficient  to  remind  the  reader 
that  the  introduction  of  this  factor  is  due  to  the  physicist's  conception  of 
the  unit  magnetic  pole  which  he  has  endued  with  the  ability  to  repel  a 
similar  imaginary  pole  with  a  force  of  1  dyne  when  the  distance  between 
the  two  unit  poles  is  1  cm.  Now,  since,  at  every  point  on  the  surface  of  a 
sphere  of  1  cm.  radius  surrounding  a  unit  magnetic  pole  placed  at  the  center, 
a  similar  pole  will  be  repelled  with  a  force  of  1  dyne,  there  must  be  unit  flux 
density  over  this  surface;  that  is  to  say,  a  flux  of  1  maxwell  per  square  centi- 
meter. The  surface  of  the  sphere  being  4?r  sq.  cm.,  it  follows  that  4ir  lines 
of  flux  must  be  thought  of  as  proceeding  from  every  pole  of  unit  strength. 
The  factor  10  in  the  denominator  converts  amperes  into  absolute  C.G.S. 
units  of  current. 


THE  MAGNETIC  CIRCUIT— ELECTROMAGNETS     13 

Magnetic  Flux. — The  unit  of  magnetic  flux  is  the  maxwell; 
it  should  be  considered  as  a  tube  of  induction  having  an  ap- 
preciable cross-section  which  may  vary  from  point  to  point.  The 
expression  "  magnetic  lines/'  which  is  customary  and  convenient, 
should  suggest  the  center  lines  of  these  small  unit  tubes  of  in- 
duction. The  total  number  of  unit  magnetic  lines  through  a 
given  cross-section  will  be  denoted  by  the  symbol  <I>. 

Flux, Density. — The  unit  of  flux  density  is  the  gauss;  it  is  a 
density  of  1  maxwell  per  centimeter  of  cross-section.  Thus,  if 
A  is  the  cross-section,  in  square  centimeters,  of  a  magnetic 
circuit  carrying  a  total  flux  of  &  maxwells  uniformly  distributed 
over  the  section,  the  flux  density  is  B  =  <J> /A  gausses.  The 
symbol  B  will  be  used  throughout  to  denote  gausses. 

Permeability. — What  may  be  thought  of  as  the  magnetic 
conductivity  of  a  substance  is  known  as  permeability  and  rep- 
resented by  the  symbol  M-  Unlike  electrical  conductivity,  it 
is  not  merely  a  physical  property  of  the  substance,  because — 
in  the  case  of  iron,  nickel,  and  cobalt — it  depends  also  upon  the 
flux  density.  For  practical  purposes,  the  permeability  of  all 
substances,  excepting  only  iron,  nickel,  and  cobalt,  is  taken  as 
unity.  Permeability  can,  therefore,  be  defined  as  the  ratio  of 
the  magnetic  conductivity  of  a  substance  to  the  magnetic  con- 
ductivity of  air. 

Reluctance  and  Permeance. — Magnetic  permeanace  is  the  re- 
ciprocal of  reluctance;  a  knowledge  of  the  permeance  of  the 
various  paths  is  useful  when  considering  magnetic  circuits  in 
parallel,  while  reluctance  is  more  convenient  to  use  when  making 
calculations  on  magnetic  paths  in  series.  The  reluctance  of  a 
path  of  unit  permeability  is  directly  proportional  to  its  length 
and  inversely  proportional  to  its  cross-section.  Thus, 

Reluctance  of  magnetic  path  in  air  =  -r 

* 

Reluctance  of  magnetic  path  in  iron  =  — j 

pA 

If  the  dimensions  are  in  centimeters,   the  reluctance  will  be 
expressed  in  oersteds. 

1  /iA 

Permeance  = — \ — 

reluctance        I 

When  calculating  reluctance  or  permeance  for  use  in  the  fun- 
damental formulas   (1)   or   (2)   it  is  important  to  express  all 


14  PRINCIPLES  OF  ELECTRICAL  DESIGN 

dimensions  in  centimeters;  no  constants  have  then  to  be  in- 
troduced because,  with  the  C.G.S.  system  of  units, 

m.m.f.  in  gilberts 


Flux  in  maxwells 


reluctance  in  oersteds 


The  sketch,  Fig.  1,  shows  a  (closed)  tube  of  induction  linked 
with  a  coil  of  wire  of  S  turns  through  which  a  current  of  /  amperes 
is  supposed  to  be  passing.  This  tube  of  induction  consists  of  a 
number  of  unit  tubes  or  so-called  magnetic  lines  each  of  which  is 
closed  on  itself.  It  follows  that  the  total  flux  3>  is  the  same 
through  all  cross-sections  of  the  magnetic  tube  of  flux  indicated 


FIG.  1.  —  Tube  of  induction  linked  with  coil. 

in  Fig.  1.  The  cross-section  may,  and  generally  does,  vary  from 
point  to  point  of  the  magnetic  circuit,  and  since  the  total  flux 
$  is  of  constant  value,  the  density  B  will  be  inversely  proportional 
to  the  cross-section.  Thus,  at  a  given  point  where  the  cross- 
section  is  A  i  square  centimeters  the  density  in  gausses  is  BI  = 


Turning  again  to  the  fundamental  formula  of  the  magnetic 
circuit,  we  have, 

m.m.f.  =  flux  X  reluctance 
or 

gilberts  =  maxwells  X  oersteds 
or 

0.4x37  =  $  X   (-7^-  +  -A-  +  etc.)  (3) 

Vd*  ^-2M2  / 


Also,  since  m.m.f.  =  magnetizing  force  X  length  of  path,  it  is 
sometimes  convenient  to  put  the  above  general  expression  in 
the  form 

QAirSI  =  HJi  +  H212  +  etc.  (4) 


THE  MAGNETIC  CIRCUIT—  ELECTROMAGNETS    15 

3.  Effect  of  Iron  in  the  Magnetic  Circuit.  —  Consider  a  toroid 
or  closed  anchor-ring  of  iron  of  uniform  cross-section  A  square 
centimeters,  wound  with  SI  ampere-turns  evenly  distributed. 

Applying  the  fundamental  formula  <f>  =  m.m.f.  X  P,  we  have, 

$  =  OAirSI  X  ^y 

in  which  I  is  the  average  length  of  the  magnetic  lines,  or  irD 
centimeters,  where  D  is  the  average  diameter  of  the  ring. 

Thus,  if  fj,  is  known,  the  flux  in  the  ring  can  be  calculated  for 
any  given  value  of  the  exciting  ampere-turns  *S7.  Since  /*  is  a 
function  of  the  density  B,  and  B  =  $/A  it  may  be  convenient  to 
put  the  above  expression  in  the  form 


B  = 

Also,  since  m.m.f.  (in  gilberts)  =  HI 

B  =  HI  X  | 

D 

whence  n  =  „'  which  explains  why  the  permeability  is  sometimes 
ti 

referred  to  as  the  multiplying  power  of  the  iron.  Thus,  for  a 
given  value  of  H,  the  magnetic  flux  in  air  will  be  H  lines  per 
square  centimeter  of  cross-section,  but  if  the  air  is  replaced  by 
iron,  it  will  be  pH  or  B  lines.  This  accounts  for  the  fact  that 
H  (the  magnetizing  force,  or  m.m.f.  per  centimeter)  is  also 
referred  to  as  the  intensity  of  the  magnetic  field,  or  magnetizing 
intensity,  and,  as  such,  expressed  in  gausses.  This  conception 
is  liable  to  lead  to  confusion  of  ideas;  but  it  is  well  to  bear  in 
mind  that,  in  air  and  other  "non-magnetic"  materials,  the 
numerical  value  of  B  is  the  same  as  that  of  H. 

For  a  given  magnetizing  force  H  (or  exciting  ampere-turns  per 
unit  length  of  circuit)  the  value  of  the  permeability,  /*,  varies 
considerably  with  different  kinds  of  iron;  it  also  depends  on  the 
past  history  of  the  particular  sample  of  iron,  and  will  not  be  the 
same  on  the  increasing  as  on  the  decreasing  curve  of  magnetiza- 
tion, as  indicated  by  the  curve  known  as  the  hysteresis  loop. 
For  the  use  of  the  designer,  careful  tests  are  usually  made  by  the 
manufacturer  on  samples  of  iron  used  in  the  construction  of 
machines,  and  curves  are  then  plotted,  or  tables  compiled,  based 
on  the  average  results  of  such  tests.  Curves  of  this  kind  have 
been  drawn  in  Figs.  2  and  3.  The  B-H  curves  of  Fig.  2  should 
be  preferred  when  the  C.G.S.  system  of  units  is  used  in  the 


16 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


xyuuu 
18000 
17000 
16000 
15000 
14000 
13000 
12000 
11000 
~  10000 

CO 

I 

^  9000 
°Q  8000 

g 

g  7000 

p 

5000 
4000 
3000 
2000 
1000 

0 

1  1 

J 

^^~ 

r-^- 

K-^- 

.  ' 

w*& 

Clg. 

—  ' 

^^ 

^s 

f*^ 

& 

c* 

^ 

X 

9^ 

/ 

/ 

/ 

/ 

/ 

/ 

1 

^^ 

^^ 

^^ 

\\ 

°^ 

^^ 

^ 

eg 

^*+ 

^ 

x* 

x^ 

x 

/ 

X 

' 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

, 

/ 

/ 

7 

/ 

/ 

0          20         40         60         80         100        120        140        160        180        200       220       240 
Magnetizing  Force  H 

FIG.  2. — B-H  curves. 


THE  MAGNETIC  CIRCUIT— ELECTROMAGNETS    17 


0* 


20         40         60         80        100       120       140       160        180       200       220       240 
Ampere-Turns  per  Inch 

FIG.  3. — Magnetization  curves  (inch  units). 


18 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


calculations;  but  so  long  as  engineers  persist  in  expressing  linear 
measurements  in  feet  and  inches,  the  curves  of  Fig.  3  will  gen- 
erally be  preferred  by  the  designer.  Fig.  4  may  be  used  for  high 
values  of  the  induction  in  armature  stampings  of  average 
quality. 

The  value  of  /*  is,  of  course,  the  ratio  between  B  of  Fig.  2  and 
the  corresponding  value  of  H,  and  curves  or  tables  giving  the 
relation  between  jj,  and  H  could  be  used;  but  it  is  generally  more 
convenient  to  read  directly  off  the  curves  of  Figs.  2,  3,  or  4,  the 


160 


£150 


130 

£120 
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I 

xllO 

E 


200  400  600  800  1000 1200  1400 1600 1800  2000  2200  2400  2600  2800  3000  3200  3400  3600  3800  4000 
Ampere  -Turns  per  Inch 

FIG.  4. — Magnetization   curve  for  armature  stampings    (high  values   of 

induction). 

flux  density  in  the  iron  coresponding  to  any  known  value  of  the 
magnetizing  force.  As  a  matter  of  fact,  it  will  be  found  that 
the  curves  are  more  frequently  used  for  the  purpose  of  determin- 
ing the  necessary  ampere-turns  to  produce  a  desired  value  of  the 
flux  density. 

4.  Magnetic  Circuits  in  Parallel. — As  an  illustration  of  the 
fundamental  relations  existing  between  magnetic  flux  and  excit- 
ing ampere-turns,  it  will  be  convenient  to  work  out  a  numerical 
example.  A  simple  case  will  be  chosen  of  magnetic  paths  in 
series  and  in  parallel,  with  small  air  gaps  in  a  circuit  consist- 
ing mainly  of  iron,  and  the  effect  of  any  leakage  flux  through 
air  paths  other  than  the  gaps  deliberately  introduced  will  be 
neglected. 


THE  MAGNETIC  CIRCUIT— ELECTROMAGNETS     19 

The  arrangement  shown  in  Fig.  5  is  supposed  t'o  represent  a 
steel  casting  consisting  of  the  magnetic  paths  (1)  and  (2)  in 
parallel,  with  the  common  path  (3)  in  series  with  them.  It 
will  be  seen  that  the  paths  (1)  and  (2)  are  provided  with  air  gaps 
and  that  the  exciting  coil  is  on  the  common  limb  (3)  only. 
Paths  (1)  and  (3)  have  iron  in  them  and  the  permeance  of  these 
paths  will  depend  upon  the  density  B  and  therefore  on  the  total 
flux  4>i  and  4>3  in  these  portions  of  the  circuit.  In  regard  to  path 
(2),  it  also  consists  mainly  of  iron,  but  the  cross-section  of  the 
iron  has  purposely  been  made  large,  so  that  the  reluctance  of 
this  path  is  practically  all  in  the  gap;  the  value  of  B  in  the  iron 
will  be  very  low,  ju  will  be  large,  and  the  reluctance  of  this  part 
of  the  iron  circuit  will  be  considered  negligible.  The  dimensions 


/^ 

A3 

=  20  sq.in. 

' 

^ 

"^ 

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1 

M 

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Jf 

*-'  0  = 

=  0.2 

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L                 j   ^ 

^              <-'  /      / 

V 

Ag  = 

iOsq. 

in. 

/ 

V 

^ 

,,-iCc 

FIG.  5. — Typical  magnetic  circuit. 

of  the  parts  are  indicated  on  the  sketch,  and  the  problem  to  be 
solved  is  the  calculation  of  the  necessary  ampere-turns  in  the  coil 
to  produce  a  given  total  flux  of,  say,  1,000,000  maxwells  through 
the  path  (1). 

The  reluctance  of  path  (1)  alone  consists  of  the  air-gap  reluc- 
tance in  series  with  the  reluctance  of  the  iron  limb  of  length  li 
and  cross-section  AI.  Thus, 

p  l°  *i 

"  Aa  X  1  "•"  Ai  X  MI 

all  dimensions  being  expressed  in  centimeters.  The  only 
unknown  quantity  is  MI,  and  this  can  be  determined  because  the 
flux  density  in  the  iron  will  be 

B'\  =  -~  -  -  =  100,000  maxwells  per  square  inch, 


20  PRINCIPLES  OF  ELECTRICAL  DESIGN 

where  the  index  "  is  added  to  the  symbol  B  to  indicate  that  inch 
units  are  used  and  that  the  density  is  not  expressed  in  gausses. 

Knowing  B  for  a  given  sample  of  iron,  the  value  of  the  per- 
meability />t  can  be  found,  and  Ri  calculated  by  putting  the 
numerical  values  in  the  above  equation.  The  necessary  m.m.f. 
for  this  portion  of  the  magnetic  circuit  (i.e.,  path  (1)  only)  is 
$1  X  Ri  gilberts. 

The  actual  procedure  would  be  simplified  by  using  the  curves 
of  Fig.  3  thus: 

Referring  to  the  upper  curve  (for  steel),  the  ampere-turns  per 
inch  required  to  produce  a  flux  density  of  100,000  lines  per  square 
inch  is  seen  to  be  80,  and  since  the  iron  portion  of  path  (l)is 
25  in.  long,  the  ampere-turns  required  to  overcome  the  reluctance 
of  iron  only  are  80  X  25  =  2,000. 

For  the  air  portion  of  path  (1),  we  have, 

m.m.f.  =  $1  X  reluctance  of  air  gap 
or 

0.4rS7  --  1,000,000  X  °4p  *  g'f  g 

whence 

SI  =  1,560 

The  total  SI  for  path  (1)  are  therefore  2,000  +  1,560  =  3,560 
or, 

m.m.f.  =  0.47T  X  3,560  =  4,470  gilberts. 

Observe  now  that  this'  m.m.f.  is  the  total  force  which  sets  up 
the  magnetic  flux  in  path  (2),  or,  in  other  words,  it  is  the  differ- 
ence of  magnetic  potential  which  produces  the  flux  of  induction 
in  the  two  parallel  paths  (1)  and  (2).  To  calculate  the  total  flux 
in  path  (2)  we  have, 

$2  =  m.m.f.  X  P2 

20  V  fi  45 
=  4,470  X    i  CO'K>I  =  227,000  maxwells 

I    /\  Z.OT: 

The  total  flux  in  limb  (3)  under  the  exciting  coil  is,  therefore, 
$3  =  ^  -f  <|>2  =  1,227,000.  The  density  in  this  core  is, 


The  necessary  ampere-turns  per  inch  (from  Fig.  3)  are  8,  and 
the  SI  for  path  (3)  are  8  X  50  =  400. 


THE  MAGNETIC  CIRCUIT— ELECTROMAGNETS    21 

The  total  ampere-turns  required  in  the  exciting  coil  to  produce 
a  flux  of  1,000,000  lines  across  the  air  gap  in  path  (1)  are  there- 
fore 3,560  -f  400  =  3,960,  which  is  the  answer  to  the  problem. 

In  all  cases  when  there  is  no  iron  in  the  magnetic  path,  i.e., 
when  n,  =  1,  as  in  the  air  gaps  of  dynamo-electric  machines,  the 
required  ampere-turns  depend  merely  on  the  density  B,  and  the 
length  I  of  the  air  gap.  The  fundamental  relation,  m.m.f.  =  HI 
can  tfren  be  written  QAirSI  =  Bl  whence, 

T> 

SI  per  centimeter  (in  air)  = 
similarly 

SI  per  inch  (in  air)  =  f      _  B  =  2.02  B 

(5) 
=  2B  approximately 

If  the  density  is  expressed  in  lines  per  square  inch, 

B" 

SI  per  inch  (in  air)  =  s~s 

O.Z 

These  formulas  are  easily  remembered  and  are  useful  for  making 
rapid  calculations. 

With  a  view  to  the  more  thorough  understanding  of  electro- 
magnetic problems  likely  to  arise  in  the  design  of  electrical 
machinery,  it  should  be  observed  that  path  (2)  of  the  magnetic 
system  shown  in  Fig.  5  may  be  thought  of  as  a  shunt — or  leakage 
—path,  the  useful  flux  being  the  1,000,000  maxwells  in  the  air 
gap  of  path  (1).  It  is  important  to  note  that  this  surplus  or 
leakage  magnetism  has  cost  nothing  to  produce,  except  in  so 
far  as  the  total  flux  <£3  is  increased  in  the  common  limb  (3), 
calling  for  a  slight  increase  in  the  necessary  exciting  ampere- 
turns.  Even  this  small  extra  I2R  loss  could  be  avoided  by  in- 
creasing the  cross-section  Az  of  the  common  limb;  but  this  would 
generally  add  to  the  cost,  not  only  because  of  the  greater  weight 
of  iron,  but  also  because  the  length  per  turn  of  the  exciting  coil 
would  usually  be  greater,  thus  increasing  the  weight  and  cost  of 
copper  if  the  PR  losses  are  to  remain  unaltered.  For  these 
reasons  alone  it  is  well  to  keep  down  the  value  of  the  leakage  flux 
in  nearly  all  designs  of  electrical  machinery;  but  the  point  here 
made  is  that  the  existence  of  a  magnetic  flux,  whether  it  be  useful 
or  leakage  magnetism,  does  not  involve  the  idea  of  loss  of  energy 
in  the  sense  of  an  PR  loss  which  must  always  be  associated  with 
the  electric  current.  Attention  is  called  to  this  matter  in  order 


22  PRINCIPLES  OF  ELECTRICAL  DESIGN 

to  emphasize  the  danger  of  carrying  too  far  the  analogy  between 
the  magnetic  and  electric  circuits.  The  product  PR  in  the 
electric  circuit  is  always  associated  with  loss  of  energy;  but 
$2  X  reluctance  does  not  represent  a  continuous  loss  of  energy  in 
the  magnetic  circuit.  If  the  energy  wasted  in  the  exciting  coil 
is  ignored,  it  may  be  said  that  the  magnetic  condition  costs 
nothing  to  maintain.  It  does,  however,  represent  a  store  of 
energy  which  has  not  been  created  without  cost;  but,  with  the 
extinction  of  the  magnetic  field,  the  whole  of  this  stored  energy  is 
given  back  to  the  electric  circuit  with  which  the  magnetic  circuit 
is  linked.  This  may  be  illustrated  by  the  analogy  of  a  frictionless 
flywheel  which  dissipates  no  energy  while  running,  but  which, 
on  being  brought  to  rest,  gives  up  all  the  energy  that  was  put 
into  it  while  being  brought  up  to  speed. 

The  dotted  lines  on  the  right-hand  side  of  the  magnetic  circuit 
shown  in  Fig.  5  indicate  two  extra  iron  paths  for  the  magnetic 
flux.  It  should  particularly  be  observed  that  the  closed  iron 
ring  D  can  be  linked  with  the  exciting  coil  as  indicated  without 
modifying  the  amount  of  the  useful  flux  through  path  (1): 
there  may  obviously  be  a  large  amount  of  flux  in  this  closed 
iron  ring,  but  it  has  cost  nothing  to  produce  because  the  exciting 
ampere-turns  have  not  been  increased.  The  same  might  be 
said  of  the  circuit  C  except  that  the  flux  in  this  circuit  has  to 
go  through  the  common  core  (3),  and  in  so  far  as  extra  ampere- 
turns  would  be  necessary  to  overcome  the  increased  reluctance 
of  the  iron  under  the  coil,  the  m.m.f.  available  for  sending  flux 
through  paths  (1)  and  (2)  would  be  reduced,  and  if  path  C 
were  of  high-grade  iron  of  large  cross-section  relatively  to  A  3, 
the  useful  flux  in  path  (1)  might  be  appreciably  reduced.  A 
proper  understanding  of  the  points  brought  out  in  the  study  of 
Fig.  5  will  greatly  facilitate  the  solution  of  practical  problems 
arising  in  the  design  of  electrical  machinery. 

5.  Calculation  of  Leakage  Paths. — The  total  amount  of  the 
magnetic  leakage  cannot  be  calculated  accurately  except  by 
making  certain  assumptions  which  are  rarely  strictly  permissible 
in  the  design  of  practical  apparatus.  Whether  the  machine  is 
an  electric  generator  or  an  electromagnet  of  the  simplest  design, 
the  useful  magnetic  flux  is  always  accompanied  by  stray  magnetic 
lines  which  do  not  follow  the  prescribed  path.  This  leakage 
flux  will  always  be  so  distributed  that  its  amount  is  a  maxi- 
mum; that  is  to  say,  the  paths  that  it  will  follow  will  always 


THE  MAGNETIC  CIRCUIT—  ELECTROMAGNETS    23 

be  such  that  the  total  permeance  of  these  leakage  paths  has 
the  greatest  possible  value.  It  is  well  to  bear  this  fact  in  mind, 
because  it  enables  the  experienced  designer  to  make  sketches  of 
various  probable  distributions  of  the  leakage  flux,  and  base  his 
calculations  on  the  arrangement  of  flux  lines  which  has  the 
greatest  permeance.  The  fact  that  the  leakage  flux  usually 
follows  air  paths  means  that  the  permeance  of  these  paths 
does  not  depend  upon  the  flux  density  B\  this  simplifies  the 
problem  because  it  is  not  necessary  to  take  into  account 
values  of  the  permeability,  /z,  other  than  unity;  the  difficulty 
lies  in  the  fact  that  —  with  the  exception  of  very  short  air 
gaps  between  relatively  large  polar  surfaces  —  it  is  rarely  possi- 
ble to  predetermine  the  distribution  of  the  stray  flux,  except 
by  making  certain  convenient  assumptions  of  questionable 
value.  .A  designer  of  experience  will  frequently  be  able  to 
estimate  flux  leakage  even  in  new  and  complicated  designs 
with  but  little  error,  and  it  is  surprising  how  the  intelligent 
application  of  empirical  or  approximate  formulas  and  rules  will 
often  conduce  to  excellent  results.  The  errors  introduced  are 
some  on  the  high  side  and  some  on  the  low  side,  and  the  averages 
are  fairly  accurate;  but  the  estimation  of  leakage  flux  —  like 
many  other  problems  to  be  solved  by  the  designer  or  practical 
engineer  —  savors  somewhat  of  scientific  guesswork;  it  calls  for 
a  combination  of  common  sense  and  engineering  judgment 
based  on  previous  experience.  The  following  examples  and 
formulas  cover  some  of  the  simplest  cases  of  flux  paths  in  air;  the 
usual  assumptions  are  made  regarding  the  paths  followed  by  the 
magnetic  lines,  but  it  may  safely  be  stated  that,  when  all  possible 
leakage  paths  have  been  considered,  and  these  formulas  applied 
to  the  calculation  of  the  leakage  flux,  the  calculated  value  will 
almost  invariably  be  something  less  than  the  actual  stray  flux 
as  subsequently  ascertained  by  experimental  means. 

Case  (a).  —  Parallel  Flat  Surfaces.  —  If  the  length  of  air  gap 
between  the  parallel  iron  surfaces  is  small  relatively  to  the 
cross-section,  and  if  the  two  surfaces  are  approximately  of  the 
same  shape  and  size,  the  average  cross-section  (see  Fig.  6)  is 


and  the  permeance  is 


p  =  f  («) 


24 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


all  dimensions  in  this  and  subsequent  examples  being  expressed 
in  centimeters. 

Case  (6). — Flat  Surfaces  of  Equal  Area  Subtending  an  Angle  6. 
—The  assumption  here  made  is  that  the  lines  of  induction  in 
the  air  gap  are  circles  described  from  a  center  on  the  axis  0 
where  the  planes  of  the  two  polar  surfaces  meet.  Let  I  =  length 
of  polar  surface  at  right  angles  to  the  plane  of  the  section  shown 
in  Fig.  7.  The  sum  of  the  permeances  of  all  the  small  paths  such 
as  dr  is  then, 

p  =  C"     I  X  dr 

J'i  '. 


(7) 


FIG.  6. — Permeance  between       FIG.  7. — Permeance  between  non- 
parallel  surfaces.  parallel  plane  surfaces. 

For  the  special  case  when  6  =  90  degrees, 


(8) 


For  the  special  case  when  0  =  180  degrees,  and  the  two  surfaces 
lie  in  the  same  plane, 

Case  (c). — Equal  Rectangular  Polar  Surfaces  in  Same  Plane. — 
This  is  a  case  similar  to  the  one  last  considered,  but  the  formula 
(9)  is  not  applicable  when  TI  is  large  relatively  to  r2  because  the 
actual  flux  lines  would  probably  be  shorter  than  the  assumed 
semicircular  paths.  With  a  greater  separation  between  the  polar 


THE  MAGNETIC  CIRCUIT—  ELECTROMAGNETS    25 


surfaces,  the  lines  of  flux  are  supposed  to  follow  the  path  indicated 
in  Fig.  8.  Let  I  stand,  as  before,  for  the  length  measured  per- 
pendicularly to  the  plane  of  the  section  shown,  then, 

IXdr 


(' 

Jo 


dr 


I   .  TTt  +  8 

logt 

7T        6<  S 


(10) 


FIG.  8. — Permeance  between  surfaces  in  same  plane. 

Case  (d). — Iron-clad  Cylindrical  Magnet. — Fig.  9  shows  a 
section  through  a  circular  magnet  such  as  might  be  used  for 
lifting  purposes.  The  exciting  coil  is  supposed  to  occupy  a 
comparatively  small  portion  of  the  total  depth,  and  in  order  to 
calculate  the  total  flux  between  the  inner  core  and  the  outer 
cylinder  forming  the  return  path  for  the  useful  flux  we  may 
consider  the  reluctance  of  the  air  path  as  being  made  up  of  a 
number  of  concentric  cylindrical  shells  of  height  h  and  thickness 
dx.  Thus, 

R        £/TJ 

reluctance  =  2    ~ — T 

M  io r  R 

The  reciprocal  of  this  quantity  is  the  permeance,  whence, 

P  =  -^J>  (ID 


When  the  radial  depth  of  the  winding  space  (R  —  r)  is  not 


26 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


greater  than  the  radius  of  the  iron  core  (r),  the  permeance  may 
be  expressed  with  sufficient  accuracy  for  practical  purposes  as 

_  mean  cross-sectional  area 

length  of  flux  path 
_  TT  (R  +  r)h 

'  (R  ~  r) 

Case  (e). — Same  as  Case  (d)  Except  that  Coils  Occupy  the 
Whole  of  the  Available  Space. — This  is  the  more  usual  case,  and 
it  is  illustrated  by  Fig.  10.  The  leakage  flux  in  the  annular 
space  occupied  by  the  windings  will  depend  not  only  upon  the 


FIG.  9. — Leakage  paths  in  circu- 
lar magnet  (space  not  occupied  by 
copper). 


r  >|<   - 


FIG.  10. — Leakage  paths  in  circu- 
lar magnet  (space  entirely  fitted  by 
exciting  coils). 


permeance  of  the  air  path,  but  also  upon  the  m.m.f.  tending  to 
establish  a  magnetic  flux.  This  m.m.f.  has  no  longer  a  constant 
value,  but,  on  the  assumption  that  the  reluctance  of  the  iron  paths 
is  negligible,  it  will  increase  according  to  a  straight-line  law  from 
zero  when  x  =  0  (see  Fig.  10)  to  a  maximum  when  x  =  h. 

Starting   with   the   fundamental   formula,    $  =  m.m.f.  X  P, 
we  have, 

~  2irdx 


=    OAwSI  X       X 


R 


whence 


or 


1  27T 

0.4mS7Xr  X 


loge  — 
r 


v     rh 

R  \  xdx 

t-Jo 


QAirSI         2nh 

T\  /\  7 


(12) 


THE  MAGNETIC  CIRCUIT—ELECTROMAGNETS    27 

which,  if  the  dimensions  are  in  centimeters,  will  be  the  leakage 
flux  in  maxwells;  and  this  is  seen  to  be  merely  the  product, 
average  value  of  m.m.f.  X  permeance. 

It  is  evident  that  this  formula  can  be  applied  to  case  (d)  in 
order  to  calculate  the  leakage  flux  in  the  space  occupied  by  the 
coil,  and  so  obtain  the  total  leakage  flux  inside  a  magnet  of  the 
type  illustrated,  where  the  coils  do  not  occupy  the  whole  of  the 
annular  air  space  between  the  core  and  the  cylinder  forming  the 
return  path. 

Case  (/). — Parallel  Cylinders. — The  permeance  of  the  air  paths 
between  the  sides  of  two  parallel  cylinders  of  diameter  d  and 


FIG.  11. — Permeance  between  parallel  cylinders. 

length  I — which  are  shown  in  section  in  Fig.  11 — cannot  be 
calculated  so  easily  as  in  the  examples  previously  considered; 
but  the  following  formula  may  be  used,1 

P  =  ~  —.  (13) 


loge 


b  +  d  -  VV  +  26d 


It  will  be  observed  that  the  logarithm  in  this  and  previous 
equations  is  to  the  base  e,  and  although  the  formula  could  be 
rewritten  to  permit  of  the  direct  use  of  tables  of  logarithms  to 
the  base  10,  there  appears  to  be  no  good  reason  for  doing  so.  If 
a  table  of  hyperbolic  logarithms  is  not  available,  the  quantity 
log(  can  always  be  obtained  by  using  a  table  of  common  logarithms 
and  multiplying  the  result  by  the  constant  2.303. 

6.  Flux  Leakage  in  Similar  Designs. — In  all  the  above  formulas 
it  will  be  seen  that  the  permeance,  P,  remains  unaltered  per 
unit  length  measured  perpendicularly  to  the  cross-section  shown 

1  This  formula  can  be  developed  mathematically  in  the  same  manner  as 
the  better-known  formulas  giving  the  electrostatic  capacity  between  parallel 
wires. 


28  PRINCIPLES  OF  ELECTRICAL  DESIGN 

in  the  sketches1  provided  the  cross-sections  are  similar,  apart 
from  the  actual  magnitude  of  the  dimensions.  Thus,  if  the 
exciting  ampere-turns  were  to  remain  constant,  the  leakage  flux  in 
similar  designs  of  apparatus  would  be  proportional  to  the  first 
power  of  the  linear  dimension  I;  but  since  the  cross-section  of 
the  winding  space  is  proportional  to  I2,  the  exciting  ampere- 
turns  would  not  remain  of  constant  value,  but  would  also  vary 
approximately  as  I2.  Given  the  same  size  of  wire  —  which 
obviates  the  necessity  of  considering  changes  in  the  winding  space 
factor  —  the  number  of  turns,  S,  will  be  proportional  to  I2, 
and  the  resistance,  R,  will  vary  as  I3.  For  the  same  rise  of 
temperature  on  the  outside  of  the  windings,  the  watts  lost  in 
heating  the  coil  must  be  proportional  to  the  cooling  surface. 
Thus, 

PR  oc  ja 


whence 
and 


81 


The  total  leakage  flux  in  similar  designs  of  magnets  will  be  pro- 
portional to  I  X  I1-5  or  I2-5,  and  as  a  rough  approximation  it 
may  be  assumed  that,  with  a  proportional  change  in  all  linear 
dimensions,  the  leakage  flux  will  vary  as  the  third  power  of  the 
linear  dimension,  or  as  the  volume  of  the  magnet. 

7.  Leakage  Coefficient.  —  The  leakage  coefficient,  or  leakage 

,.    useful  flux  +  leakage  flux 
factor,  is  the  ratio  -  .  1  a  — 

useful  flux 

or 


where  <J>j  is  the  total  number  of  leakage  lines  calculated  for  every 
path  where  an  appreciable  amoun-t  of  leakage  is  likely  to  occur. 
When  designing  electromagnets  or  the  frames  of  dynamo 
machines,  a  fairly  close  estimate  of  the  probable  leakage  factor 
is  necessary  in  order  to  be  sure  that  sufficient  iron  section  will  be 

1  The  sections  shown  in  Figs.  9  and  10  have,  for  convenience,  been  taken 
through  the  axis  of  length  instead  of  at  right  angles  to  this  axis  as  in  the 
other  examples. 


THE  MAGNETIC  CIRCUIT—  ELECTROMAGNETS    29 

provided  under  the  magnetizing  coils  and  in  the  yoke  to  carry  the 
leakage  flux  in  addition  to  the  useful  flux.  The  leakage  factor 
is  always  greater  than  unity,  and  the  product  of  the  useful  flux 
by  the  leakage  factor  will  be  the  total  flux  to  be  carried  by 
certain  portions  of  the  magnetic  circuit  enclosed  by  the  exciting 
coils. 

8.  Tractive  Force.  —  The  tractive  effort,  or  the  tension  which 
exists  along  the  magnetic  lines  of  force,  is  one  of  the  effects  of 
magnetism  which  it  is  necessary  to  calculate,  not  only  in  electro- 
magnets for  lifting  purposes,  or  in  magnetic  clutches  or  brakes, 
where  this  is  the  most  important  function  of  the  magnetism; 
but  also  in  rotating  electric  machinery,  where  decentralization  of 
the  rotating  parts  may  lead  to  very  serious  results  owing  to  the 
unbalancing  of  the  magnetic  pull. 

MAXWELL'S  formula  is, 

B2 

Force  in  dynes  =  0  -A  (14) 

O7T 

where  A  is  the  cross-section  in  square  centimeters  of  a  given  area 
over  which  the  flux  density,  B,  is  assumed  to  have  a  constant 
value. 

This  formula  can  be  used  to  calculate  the  pull  between  two 
parallel  polar  surfaces  when  the  air  gap  between  them  is  small 
relatively  to  the  area  of  the  surfaces.  The  engineer  desires  to 
know  the  pull  in  pounds  exerted  between  the  two  surfaces,  and 
since  1  Ib.  =  444,800  dynes,  the  above  formula  can  be  written, 

B* 

putt,  in  pounds  per  square  centimeter  =  1  1  1  Qn  -__        (15) 

11, 


or 

B2 
pull,  in  pounds  per  square  inch  =  Y73Q  000 

In  both  of  these  formulas  the  density,  B,  is  expressed  in  gausses 
(i.e.,  in  C.G.S.  lines  per  square  centimeter). 
If  B)f  stands  for  lines  per  square  inch,  then, 


pull,  in  pounds  per  square  inch 


B2 


72  v  10« 

If  the  density  is  not  constant  over  the  whole  surface  considered, 
the  area  must  be  divided  into  small  sections,  after  which  a 
summation  of  the  component  forces  can  be  made.  In  averaging 
the  density  to  get  a  mean  result,  it  is  obviously  not  the  square  of 
the  average  density  that  must  be  taken,  but  the  average  of  the 


30 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


squares  of  the  densities  taken  over  the  various  component  areas 
of  the  cross-section  considered.  This  is  briefly  summed  up  in 
the  general  expression, 

pull,  in  dynes  =  ^-  I  B2dA 

Pull  between  Inclined  Surfaces. — Conical  Plungers. — Sketch 
(a)  of  Fig.  12  shows  a  portion  of  an  electromagnet  of  rectangular 
cross-section,  with  air  gap  (of  length  I)  normal  to  the  direction 
of  movement.  The  sketch  (b)  shows  a  similar  bar  of  iron,  but 
with  the  air  gap  inclined  at  an  angle  6  with  the  normal  cross- 
section.  The  total  movement,  which  is  supposed  to  be  confined 
to  the  direction  parallel  to  the  length  of  the  bar,  is  the  same  in 
both  cases;  that  is  to  say,  the  air  gap  measured  in  the  direction 


<*)  v, 

FIG.  12. — Magnet  with  inclined  air  gap. 

of  motion,  has  the  same  value,  I,  although  the  actual  air  gap 
measured  normally  to  the  polar  surfaces  is  smaller  in  (b)  than 
in  (a) .  The  magnetic  pull  will  actually  be  exerted  in  a  direction 
normal  to  the  opposing  surfaces,  that  is  to  say,  in  the  direction 
OF8  in  case  (b),  although  it  is  the  mechanical  force  exerted  in 
the  direction  OFz  which  it  is  proposed  to  calculate. 

For  the  perpendicular  gap  (sketch  a)  we  can  write, 

total  longitudinal   force  Fl  =  kBl2Al  (18) 

where  k  is  a  constant. 

For  the  inclined  gap  (sketch  6) ; 

total  longitudinal  force  F2  =  fc#22A2  X  cos  6  (19) 
Now  express  equation  (19)  in  terms  of  AI  and  Bi.  With  the 
ampere-turns  of  constant  value,  and  considering  the  reluctance 
of  the  air  gap  only,  the  flux  density  will  be  inversely  proportional 
to  the  shortest  distance  between  the  two  parallel  surfaces. 
Thus, 


1 
Zcos  6 


THE  MAGNETIC  CIRCUIT— ELECTROMAGNETS    31 


therefore 
Also,  since 


Bl 

cos  0 


A2  = 


cos  0 


A! 

cos  0 


X  cos0 


cos2 "0"    ~  cos2  0  (20) 

The  same  relation  holds  good  for  cone-shaped  pole  pieces. 
Thus,  referring  to  Fig.  13,  in  which  the  magnet  core  is  supposed 
to  have  a  circular  cross-section  of  radius  r; 

FI  =  kBi2Ai  =  kBi2irr2  for  normal  gap 


and 


FIG.   13. — Magnet  with  conical  pole  faces. 


F2  =  kBzzAz  X  cos  0 

for  conical  gap;  where  the  factor  cos  0  is  introduced  as  before 
to  obtain  the  axial  component  of  the  magnetic  forces. 

The  conical  surface,  which  corresponds  to  the  cross-sectional 
area  of  the  air  gap,  is, 

A*  =   y2X2wrX  ^  =  —e  =  ^~ 
Also,  for  the  same  exciting  ampere-turns,  we  have  as  before, 


I  cos  0 


and, 


COS20 


which  is  identical  with  formula  (20). 


32  PRINCIPLES  OF  ELECTRICAL  DESIGN 

Thus  in  both  cases  the  inclined  gap  has  the  effect  of  increasing 
the  initial  pull  for  the  same  total  length  of  travel.  This  is 
sometimes  an  advantage,  and  conical  poles  are  occasionally 
introduced  in  designs  of  electromagnets  when  the  total  travel  is 
small  relatively  to  the  diameter  of  the  plunger.  In  this  manner 
it  is  possible  to  obtain  increased  length  of  travel  without  adding 
to  the  weight  of  the  magnet.  For  the  same  initial  pull,  the 
length  of  travel  obtainable  by  providing  conical  surfaces  is, 

li 
2  ~  cos2  0 

where  l\  is  the  length  of  the  normal  gap  which  corresponds  to 
the  required  initial  pull.  This  formula  is  easily  derived  from 
the  expressions  previously  developed. 

There  is  a  limit  to  the  amount  of  taper  that  can  be  put  on  the 
conical  pole  pieces,  and  a  large  amount  of  taper  will  prove  to  be 
of  little  use.  It  should  be  noted  that  a  limit  of  usefulness  is 
reached  when  the  flux  density  in  the  iron  core  approaches  satura- 
tion limits,  because  the  air-gap  density — which  determines  the 
magnetic  pull — cannot  then  be  carried  up  to  high  values,  even 
with  greatly  increased  exciting  ampere-turns.  A  reference  to  the 
curves  of  Figs.  2  or  3  (pages  16,  17)  will  enable  the  designer  to 
judge  when  the  density  in  the  magnet  is  approaching  uneconom- 
ical values.  Thus,  in  the  case  of  cast  iron  it  will  rarely  pay  to 
carry  the  induction  above  11,000  gausses,  while,  in  wrought  iron, 
or  cast  steel  as  used  for  electronlagnets,1  the  upper  limit  may  be 
placed  at  about  19,000  gausses,  although,  as  will  be  explained 
later,  it  is  often  advantageous  to  force  the  density  up  to  higher 
values  in  the  teeth  of  laminated  armature  cores. 

9.  Materials — Wire  and  Insulation. — Before  going  further  into 
the  design  of  electromagnets  it  will  be  advisable  to  consider 
briefly  the  qualities  of  the  materials  used  in  their  construction. 
The  most  important  of  these  materials  is  the  iron,  which  con- 
centrates the  magnetic  flux  and  so  provides  the  necessary  dis- 
tribution and  density  in  the  air  gap  where  it  performs  the  duty 
required  of  it.  The  effect  of  iron  in  the  magnetic  circuit  has 
however  already  been  discussed  at  some  length,  and  as  its  various 
properties  will  be  considered  further  in  the  course  of  subsequent 
articles,  it  is  proposed  to  confine  the  remarks  immediately  fol- 

1  This  is  practically  pure  iron,  with  magnetic  characteristics  very  similar 
to  those  of  soft  annealed  iron  of  good  quality. 


THE  MAGNETIC  CIRCUIT— ELECTROMAGNETS    33 

lowing  to  the  only  other  materials  of  consequence  in  the  design  of 
magnets  or  dynamos,  namely,  the  copper  wire,  which  is  the 
material  universally  used  for  the  windings,  and  the  insulating 
materials,  which  prevent  electrical  contact  between  neighboring 
turns  of  wire,  and  also  between  the  winding  as  a  whole  and  the 
iron  of  the  magnetic  circuit  or  supporting  framework. 

For  the  operation  of  electromagnets,  high  voltages  are  rarely 
used,  and  the  provision  of  appropriate  insulation  presents  no 
serious  difficulties;  but  it  must  not  be  overlooked  that,  when 
the  inductance  is  great — i.e.,  when  the  flux  links  with  a  large 
number  of  turns  and  the  product  maxwells  X  number  of  turns 
is  large — there  may  be,  at  the  instant  of  switching  off  the  current, 
differences  of  potential  between  neighboring  turns  of  wire, 
considerably  in  excess  of  the  normal  potential  difference  cal- 
culated on  the  assumption  of  a  steady  impressed  voltage  between 
the  terminals  of  the  coil.  In  the  design  of  continuous-current 
machines,  pressures  up  to  5,000  volts  may  have  to  be  considered, 
and  in  alternating-current  generators,  the  pressure  may  be  as  high 
as,  but  rarely  in  excess  of,  16,000  volts.  The  higher  pressures, 
as  used  for  transmission  of  energy  to  great  distances,  are  obtained 
by  means  of  static  transformers,  and  the  question  of  insulation 
then  becomes  of  such  great  importance  that  it  has  to  be  very 
thoroughly  studied  by  experts.  Pressures  of  100,000  volts  are 
now  common  for  step-up  transformers,  and  there  are  many 
transformers  actually  in  operation  at  150,000  volts  and  even 
higher  pressures;  so  that  the  provision  of  the  requisite  insulation 
for  machines  working  at  pressures  not  exceeding  16,000  volts 
(which  is  the  limit  for  any  of  the  designs  dealt  with  in  this  book) 
offers  no  insuperable  difficulties.  It  is,  therefore,  proposed  to 
devote  but  little  space  to  the  discussion  of  insulation  problems; 
although,  as  occasion  arises,  data  and  information  of  a  practical 
nature  will  be  given. 

Copper  Wire. — With  silver  as  the  one  exception,  copper  is 
the  metal  with  the  highest  electrical  conductivity;  it  is  also 
mechanically  strong,  easy  to  handle,  and  generally  the  most 
suitable  material  for  electrical  windings.  The  resistance  of  a 
given  size  and  length  of  wire  is  usually  obtained  by  reference 
to  a  wire  table,  similar  to  the  accompanying  tables,  which  con- 
tain such  information  as  the  designer  of  electrical  apparatus 
requires.  The  very  large,  and  the  very  small,  sizes  of  wire  are 
omitted;  but  wire  tables  for  the  use  of  electrical  engineers  are 

3 


34 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


so  common  in  electrical  handbooks  and  textbooks,  that  par- 
ticulars of  sizes  not  here  included  can  generally  be  obtained 
without  difficulty. 

The  Brown  and  Sharp  gage  is  commonly  used  in  America, 
while  the  legal  standard  gage  (S.W.G.) — which  has  been  adopted 
by  the  Engineering  Standards  Committe — is  used  almost  with- 
out exception  by  electrical  engineers  in  England.  In  using  the 
accompanying  tables,  reference  should  always  be  made  to  the 
heading,  to  ensure  that  the  figures  relate  to  the  required  wire 
gage. 

WIRE  TABLE,  BROWN  AND  SHARP  GAGE,  COPPER 


Gage 
No., 
B.  &S. 

Diameter, 
inches 
(bare) 

Area    of    cross- 
section 

Weight, 
Ib.  per 
1,000  ft. 
(bare) 

Approx. 
diame- 
ter 
D.C.C. 

(mils) 

Approx. 
number 
of  turns 
per  inch 
D.C.C. 

Resistance,  ohms 
per  1,000  ft.i 

Gage 
No., 
B.  &S. 

Square 
inches 

Circu- 
lar mils 

15°C. 

(59°F.) 

60°C. 

(140°F.) 

0 

0.3249 

0.08291 

105,560 

319.5 

338 

2.95 

0.0964 

0.1142 

0 

1 
2 

0.2893 
0.2576 

0.06573 
0.05212 

83,690 
66,370 

253.3 
200.9 

302 
270 

3.30 
3.69 

0.1217 
0.1534 

0.1440 
0.1816 

1 
2 

3 

0.2294 

0.04133 

52,630 

159.3 

242 

4*12 

0.1934 

0  .  2290 

3 

4 

0.2043 

0.03278 

41,740 

126.4 

216 

4.60 

0.2439 

0  .  2888 

4 

5 

0.1819 

0  .  02600 

33,090 

101.2 

194 

5.13 

0.3076 

0.3642 

5 

6 

0.1620 

0.02061 

26,250 

79.5 

174 

5.70 

0.388 

0.459 

6 

7 

0.1443 

0.01635 

20,820 

63.0 

156 

6.36 

0.489 

0.579 

7 

8 

0.1285 

0.01297 

16,510 

50.0 

140 

7.10 

0.617 

0.730 

8 

9 

0.1144 

0.01028 

13,090 

39.6 

126 

7.88 

0.778 

0.921 

9 

10 

0.1019 

0.00815 

10,380 

31.4 

114 

8.70 

0.981 

1.161 

10 

11 

0.0907 

0.00646 

8,230 

24.9 

103 

9.60 

1.237 

1.464 

11 

12 

0  .  0808 

0.00513 

6,530 

19.8 

93 

10.65 

1.559 

1.846 

12 

13 

0.0720 

0  .  00407 

5,178 

15.7 

84 

11.80 

1.966 

2.328 

13 

14 

0.0641 

0.00323 

4,107 

12.43 

76 

13.0 

2.480 

2.936 

14 

15 

0.0571 

0.00256 

3,260 

9.86 

68 

14.5 

3.127 

3.702 

15 

16 

0  .  0508 

0.00203 

2,583 

7.82 

62 

15.9 

3.942 

4.667 

16 

17 

0.0453 

0.00161 

2,048 

6.20 

56 

17.5 

4.973 

5.887 

17 

18 

0  .  0403 

0.001276 

1,624 

4.92 

51 

19.2 

6.27 

7.42 

18 

19 

0.0359 

0.001012 

1,288 

3.90 

46 

21.3 

7.90 

9.36 

19 

20 

0.0320 

0  .  000802 

1,022 

3.09 

42 

23.3 

9.97 

11.80 

20 

21 

0.0285 

0  .  000636 

810 

2.45 

38 

25.6 

12.57 

14.88 

21 

22 

0.0253 

0  .  000503 

642 

1.945 

35 

27.8 

15.86 

18.77 

22 

23 

0.0226 

0.000401 

510 

1.542 

32 

30.3 

20.00 

23.66 

23 

24 

0.0201 

0.000317 

404 

1.223 

30 

32.3 

25.20 

29.84 

24 

25 

0.0179 

0.000252 

320 

0.970 

27 

35.7 

31.80 

37.60 

25 

26 

0.0159 

0.0001985 

254 

0.769 

24 

40.0 

40.20 

47.50 

26 

27 

0.0142 

0.0001584 

202 

0.610 

22 

43.5 

50.60 

60.00 

27 

28 

0.0126 

0.0001247 

159 

0.484 

21 

45.5 

63.80 

75.40 

28 

1  A  variation  in  resistance  up  to  2  per  cent,  increase  on  the  calculated  values  for  pure 
copper  is  generally  allowed. 


THE  MAGNETIC  CIRCUIT— ELECTROMAGNETS    35 


WIRE  TABLE,  STANDARD  WIRE  GAGE,  COPPER 


Gage 
No., 
S.W.G. 

Diameter, 
inches, 
(bare) 

Area    of    cross- 
section 

Weight, 
Ib.  per 
1,000ft. 
(bare) 

Appro*.  |  Appro*, 
diame-    number 
ter       of  turns 
D.C.C.  per  inch 
(mils)       D.C.C. 

Resistance,   ohms 
per  1,000ft.1 

Gage 
No., 
S.W.G. 

Square 
inches 

Circu- 
lar '  mils 

15°C. 
(59°F.) 

60°C. 
(140°F.) 

0 

0.324 

0.08245 

105,000 

318.0 

340 

2.93 

0.097 

0.1148 

0 

1 

0.300 

0.07070 

90,000272.0 

316 

3.15 

0.1131 

0.1339 

1 

2 

0.276 

0  .  05980 

76,180231.0 

292 

3.41 

0.1337 

0.1583 

2 

3 

0.252 

0  .  05000 

63,500  192.0 

268 

3.72 

0.1604 

0.1900 

3 

4 

0.232 

0.04230 

57,820 

166.0 

248 

4.01 

0.1892 

0.2240 

4 

5 

0.212 

0.03530 

44,940 

136.0 

228 

4.37 

0  .  2260 

ii  _><;s:; 

5 

6 

0.192 

0  .  02895 

36,860 

111.5 

208 

4.78 

0.2767 

0.3276 

6 

•7 

0.176 

0.02433 

30,980 

93.8 

192 

5.18 

0.3291 

0  .  3896 

7 

8 

0.160 

0.02010 

25.600 

77.5 

174 

5.72 

0.398 

0.471 

8 

9 

0.144 

0.01630 

20,740 

62.8 

158 

6.28 

0.491 

0.581 

9 

10 

0.128 

0.01287 

16,380 

49.6 

140 

7.10 

0.625 

0.740 

10 

11 

0.116 

0.01057 

13,460 

40.7 

128 

7.75 

0.759 

0.902 

11 

12 

0.104 

0.00850 

10,820 

32.7 

116 

8.55 

0.941 

1.114 

12 

13 

0  092 

0.00665 

8,465 

25.6 

104 

9.52 

1.203 

1.424 

13 

14 

0.080 

0.00503 

6.400 

19.4 

92 

10.75 

1.591 

1.884 

14 

15 

0.072 

0.00407 

5,185 

15.7 

84 

11.75 

1.964 

2.325 

15 

16 

0.064 

0.00322 

4,095 

12.4 

76 

13.0 

2.486 

2.943 

16 

17 

0.056 

0.00246 

3,135 

9.5 

68 

14.5 

3.246 

3.844 

17 

18 

0.048 

0.00181 

2,305 

7.0- 

58 

17.0 

4.420 

5.234 

18 

19 

0.040 

0.001257 

1,600 

4.84 

50 

19.6 

6.37 

7.54 

19 

20 

0.036 

0.001018 

1,296 

3.92 

46 

21.3 

7.85 

9.30 

20 

21 

0.032 

0.000804 

1,024 

3.10 

42 

23.2 

9.94 

11.77 

21 

22 

0.028 

0.000616 

784 

2.37 

38 

25.6 

12.99 

15.38 

22 

23 

0.024 

0.000452 

576 

1.74 

34 

28.6 

17.67 

20.93 

23 

24 

0.022 

0.000380 

484 

1.47 

31 

31.0 

21.08 

24.91 

24 

25 

0.020 

0.000314 

400 

1.21 

29 

33.0 

25.50 

30.20 

25 

26 

0.018 

0.000255 

324 

0.98 

27 

36.0 

31.40 

37.20 

26 

27 

0.0164 

0.000211 

270 

0.81 

25 

38.0 

37.80 

44.70 

27 

28 

0.0148 

0.000172 

219 

0.665 

24 

40.0 

46.50 

55.00 

28 

1  A  variation  in  resistance  up  to  2  per  cent,  increase  on  the  calculated  values  for  pure 
copper  is  generally  allowed. 

The  cross-section  of  a  wire  may  be  expressed  in  square  inches 
or  in  square  mils  (1  mil  =  1/1000  in.) ;  the  metric  system  is  rarely 
used  in  English-speaking  countries. 

Circular  Mils. — The  cross-section  of  a  wire  or  conductor  may 
also  be  expressed  in  "  circular  mils."  This  is  the  unit  of  area 
commonly  used  in  America  when  the  cross-section  of  electrical 
conductors  is  referred  to.  The  confusion  of  ideas  resulting  from 
the  conception  of  the  circular  mil  as  a  unit  of  area  may  be  com- 
pensated for  by  certain  practical  advantages,  but  these  advantages 
are  not  obvious.  The  circular  mil  is  the  area  of  a  circle  1  mil 


36  PRINCIPLES  OF  ELECTRICAL  DESIGN 

in  diameter,  and  the  number  of  circular  mils  in  a  given  area 

is  therefore  greater  than  the  number  of  square  mils.     Thus,  in  1 

4 
sq.  in.  there  are  1,000,000  square  mils;  but  106  X      =  1,273,237 

7T 

circular  mils.  The  cross-section  of  a  cylindrical  wire  in  circular 
mils  is, 

(m)  =  (diameter  in  mils)2 

true  area  in  square  mils 
0.7854 

The  area  of  any  conductor  expressed  in  circular  mils  is  always 
greater  than  the  true  area  expressed  in  square  mils. 

Simple  Formulas  for  Resistance  of  Wires. — A  very  convenient 
and  easily  remembered  rule  is  that  the  resistance  of  any  copper 
wire1  is  1  ohm  per  circular  mil  per  inch  length,  or 

I" 

R  =   T^T  (21) 

(m) 

at  a  temperature  of  about  60°C.  (or  140°F.).  This  formula  is 
therefore  applicable  to  the  calculation  of  coil  resistances  under 
operating  conditions,  when  they  are  hot. 

The  system  on  which  the  B.  &  S.  (Brown  and  Sharp)  gage  is 
based,  exactly  halves  the  cross-section  with  an  increase  of  three 
sizes.  It  will  also  be  found  that  a  No.  10  B.  &  S.  copper  wire 
has  a  cross-section  of  about  10,000  circular  mils  (diameter  =  0.1 
in.  approx.)  and  its  resistance  at  normal  temperatures  (about 
20°C.)  is  1  ohm  per  1,000  ft.  Thus,  for  approximate  calculations, 
sizes  of  wire  on  the  B.  &  S.  gage  can  be  determined  if  necessary 
without  reference  to  tables. 

The  weight  of  any  size  of  round  copper  wire  may  be  calculated 
by  the  formula: 

d2 

Weight  in  pounds  per  1,000  ft.  =  ^7^  (22) 

ooU 

where  d  =  diameter  in  mils. 

Variation  of  Resistance  with  Temperature. — If  the  resistance  of 
a  wire  is  known  for  any  given  temperature  it  can  readily  be 
calculated  for  any  other  temperature  by  remembering  that  the 
resistance  of  all  pure  metals  tends  to  become  zero  at  the  absolute 
zero  of  temperature,  and  that  the  variations  in  resistance  follow 

1  The  specific  resistance  of  commercial  wires  can  be,  and  usually  is,  equal 
to  that  of  pure  electrolytic  copper  of  100  per  cent,  conductivity  by 
Matthiesson's  standard. 


THE  MAGNETIC   CIRCUIT— ELECTROMAGNETS    37 


a  straight-line  law,  all  as  indicated  in  Fig.  14.  Thus,  if  Rt  and 
RQ  stand  respectively  for  the  resistances  at  temperatures  of  t 
degrees  and  zero  degrees,  the  relation  is, 


Rt  =  RQ(1  +  at) 


(23) 


If  it  is  desired  to  calculate  the 
change  in  resistance  which  occurs 
when  the  temperature  is  raised 
from  ti  to  t2  degrees,  we  have, 

R2  =  RQ(l  -f  at2) 

Dividing  the  first  equation  by 
the  second,  in  order  to  eliminate 
R0j  we  get, 

(1  +  at,) 
(1  +  at,)  Kl 


If, 


(24)  -^3  c  - 


Resistance 


(Absolute  Zero) 


FIG.     14.—  Diagram     illustrating 
by  which  the  resistance  R2  at  the    variation  of  resistance  with  temper- 

temperature  tz  can  be  calculated 

when  the  resistance  R\  at  the  temperature  t\  is  known. 

The  coefficient  a  =  0.004  if  the  temperatures  are  expressed  in 
degrees  Centigrade.  If  temperatures  are  read  on  the  Fahrenheit 
scale,  a  =  0.0024. 

Numerical  Example  —  Change  of  Resistance  with  Temperature.— 
The  resistance  of  copper  per  circular  mil  per  foot  is  12  ohms  at 
60°C.  Calculate  the  temperature  at  which  the  resistance  will 
be  10  ohms  per  circular  mil  per  foot. 


60a) 

Rt  =  R0(l  +  ta) 

Divide  the  first  equation  by  the  second,  and  solve  for  t,  the  value 
of  which  is  found  to  be, 

Rt(l  +  60a)  -  fleo 


t  = 


X  a 


Substitute  the  numerical  values,  R  6o  =  12;  Rt  =  10;  and  a  = 
0.004  which  will  give  the  answer  8.33°C. 

Insulating  Materials.  —  The  covering  on  the  copper  wires  may 
consist  of  one,  two,  or  three  layers  of  cotton  or  silk.  Silk  cover- 
ings are  used  only  on  the  smaller  sizes,  especially  when  it  is 
important  to  economize  space,  that  is  to  say,  where  the  space 


38 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


taken  up  by  the  cotton  covering  would  be  excessive  in  propor- 
tion to  the  cross-section  of  copper.  When  great  economy  of 
space  is  necessary,  enamelled  wire  may  be  used.  This  is  simply 
bare  copper  wire  on  which  a  thin  coating  of  flexible  enamel  has 
been  applied  by  a  special  process.  Enamelled  wire  may  often 
be  used  to  advantage,  especially  in  connection  with  the  very 
small  diameters;  but  there  is  always  the  possibility  of  contact 
between  adjacent  wires  at  abnormally  high  temperatures,  and  in 

BELDENAMEL  WIRE  DATA 


Nos.  B.  &  S. 
gage 

B.  &  S.  sizes,  bare 
(inches) 

Increase  thickness 
of  enamel  insula- 
tion 

Allowable  varia- 
tion in  thickness 

Average  diameter 
over  enamel 

13 

0.0720 

0.002 

0.0005 

0.074 

14 

0  .  0641 

0.002 

0.0005 

0.0661 

15 

0.0571 

0.002 

0.0005 

0.0591 

16 

0.0508 

0.002 

0.0005 

0.0528 

17 

0.0452 

0.0018 

0.0004 

0.047 

18 

0.0403 

0.0018 

0.0004 

0.0421 

19 

0.0359 

0.0018 

0.0004 

0.0377 

20 

0.0320 

0.0018 

0.0004 

0.0338 

21 

0.0284 

0.0017 

0.0004 

0.0301 

22 

0.0253 

0.0016 

0.0004 

0.0269 

23 

0.0225 

0.0015 

0.0004 

0.0240 

24 

0.0201 

0.0014 

0.0003 

0.0215 

25 

0.0179 

0.0013 

0.0003 

0.0192 

26 

0.0159 

0.0012 

0.0003 

0.0171 

27 

0.0142 

0.0011 

0.0003 

0.0153 

28 

0.0126 

0.0010 

0.0003 

0.0136 

29 

0.0112 

0.0009 

0.0003 

0.0121 

30 

0.0100 

0.0008 

0.0002 

0.0108 

31 

0.0089 

0.0008 

0.0002 

0.0097 

32 

0.0079 

0.0007 

0  .  0002 

0.0086 

33 

0.0071 

0.0007 

0.0002 

0.0078 

34 

0.0063 

0.0006 

0.0002 

0.0069 

35 

0.0056 

0.0006 

0.0001 

0.0062 

36 

0.005 

0.0005 

0.0001 

0.0055 

37 

0.0044 

0.0005 

0.0001 

0.0049 

38 

0.004 

0.0004 

0.0001 

0.0044 

39 

0.0035 

0.0004 

0.0001 

0.0039 

40 

0.0031 

0.0004 

0.0001 

0.0035 

THE  MAGNETIC  CIRCUIT— ELECTROMAGNETS    39 

many  cases  enamelled  wire  protected  by  a  single  covering  of 
cotton  has  been  used  with  very  satisfactory  results.  In  all  cases 
when  it  is  desired  to  save  space  by  reducing  the  thickness  of 
insulation  on  wires,  the  points  to  be  considered  are:  (1)  insulation; 
(2)  durability;  and  (3)  cost.  The  price  of  the  silk  covering  is  of 
course  much  higher  than  that  of  the  cotton  covering. 

Enamel  insulation  does  not  add  much  to  the  diameter  of  the 
wire  as  will  be  seen  by  reference  to  the  accompanying  table  based 
on  data  kindly  furnished  by  the  Belden  Manufacturing  Co.  of 
Chicago.  This  wire  will  not  suffer  injury  with  the  temperature 
maintained  at  200°F.  continuously,  and  it  will  withstand  without 
breakdown  a  pressure  of  900  volts  per  mil  thickness  of  enamel; 
but  on  account  of  the  possibility  of  abrasion  during  winding,  a 
large  factor  of  safety  (not  less  than  four)  should  be  used,  and 
indeed  it  is  always  advisable  to  place  paper  between  the  layers  of 
enamelled  wire,  unless  a  careful  study  of  the  conditions  appears 
to  justify  its  omission. 

Triple  cotton  covering  can  be  used  with  advantage  on  the 
larger  sizes  of  wire  when  the  working  pressure  between  adjacent 
turns  exceeds  20  volts.  When  extra  insulation  is  required  be- 
tween the  layers  of  the  winding,  this  is  usually  provided  in  the 
form  of  one  or  more  thicknesses  of  paper  or  varnished  cloth. 
It  is  the  insulation  between  the  finishing  turns  of  a  layer  of  wire 
and  the  winding  immediately  below  which  requires  special  at- 
tention, because  this  is  where  the  difference  of  potential  is  great- 
est. One  advantage  of  the  ordinary  cotton  covering  is  that  it 
lends  itself  admirably  to  treatment  with  oil  or  varnish,  either 
before  or  after  winding. 

Space  Factor. — The  amount  of  space  taken  up  by  the  insulation 
and  the  air  pockets  between  wires  of  circular  cross-section  is 
important,  because  it  reduces  the  cross-section  of  copper  in  the 

coil.     If  A  is  the  cross-section  of  the  copper,  and  A'  the  total  area 

j^ 

of  cross-section  through  the  winding,  the  ratio  -p  is  called  the 

space  factor.  The  calculated  space  factor,  based  on  the  assump- 
tion of  a  known  diameter  over  the  insulation,  and  a  close  packing 
of  the  wires,  does  not  always  agree  with  the  value  obtained  in 
practice,  but  the  curves  of  Fig.  15  will  be  found  to  give  good 
average  values.  It  will  be  understood  that  the  space  factors  of 
Fig.  15  include  no  allowance  for  extra  insulation  between  layers 
of  wire  or  for  the  necessary  lining  of  the  spool  upon  which  the  coil 


40 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


is  wound.  If  the  number  of  turns  per  layer  is  very  small,  there 
will  be  an  appreciable  loss  of  space  due  to  the  turning  back  of  the 
wire  at  the  end  of  each  layer. 

Insulation  on  Spools  or  Metal  Forms. — The  materials  used  for 
insulating  between  the  winding  as  a  whole  and  any  grounded 
metal  by  which  it  is  supported  include  mica,  micanite  paper  and 
cloth,  pressboard,  "presspahn,"  varnished  cambric,  oiled  linen 


U.72 

0  70 

\ 

\ 

svv^ 

fl  fiR 

\ 

\ 

0  66 

\ 

sx 

^ 

>n» 

0  64 

\ 

^ 

*r* 

fl  fi9 

\« 

rt 

^ 

9> 

\ 

«  n  fin 

§ 

fc 

\ 

\ 

Jo  *y? 

* 

s 

\ 

n  "y? 

s 

\ 

\ 

g°-°0 

'~  n   KA 

\ 

.£ 

\ 

0  ^Q 

\ 

0  48 

\ 

0  4fi 

\ 

0  44 

\ 

6       8 


32     34 


10     12     14      16     18     20      22     24     26     28 
Size  of  Wire,  B  &  S  Gauge 

FIG.  15. — Space  factors  for  wires  of  circular  cross-section. 

or  cotton  (empire  cloth),  cotton  tape,  etc.  The  voltage  that 
some  of  these  materials  will  withstand  before  breakdown  is 
approximately  as  follows :  empire  cloth  (usually  7  to  8  mils  thick) 
will  rarely  puncture  with  less  than  600  volts  per  mil;  mica  will 
withstand  about  800  volts  per  mil;  and  micanite  paper  or  cloth 
— which  affords  also  an  excellent  mechanical  protection — can 
generally  be  relied  on  to  withstand  400  volts  per  mil.  A  large 
factor  of  safety  is  usually  allowed,  especially  on  the  lower 
voltages.  With  a  good  quality  of  insulation,  the  total  thickness 
between  the  cotton-covered  wires  and  the  supporting  metal  work 
should  have  the  following  values: 


THE  MAGNETIC  CIRCUIT— ELECTROMAGNETS    41 


Up  to  500  volts 0.045  in. 

For  1,000  volts 0.060  in. 

For  2,000  volts 0.080  in. 

For  3,000  volts 0.10     in. 

For  higher  pressures,  up  to  12,000  volts,  add  0.03  in.  per  1,000 
volts  increase. 

10.  Calculation  of  Magnet  Windings. — The  calculation  of  the 
ampere-turns  necessary  to  produce  a  given  flux  of  magnetism 
has  already  been  explained  (see  Arts.  3  and  4),  and  it  is  a  fairly 
simple  matter  to  determine  the 
exciting  force  approximately, 
provided  the  magnetic  circuit 
consists  mainly  of  iron  of  known 
magnetic  characteristics,  and 
that  the  air  gaps  are  short. 
These  calculations  will  be  more 
fully  illustrated  when  working 

out  one  or  two  numerical  exam-  t 

.      .     FIG.   16. — Cylindrical  magnet  coil, 
pies;   but  for  the  present  it  is 

assumed  that  a  definite  number  of  ampere-turns,  SI,  have  to  be 
wound  on  a  bobbin  or  former,  and  that  the  applied  D.C.  potential 
difference,  E,  is  known. 

If  SI  =  the  total  ampere-turns  in  the  coil  shown  in  Fig.  16, 
then,  whatever  may  be  the  number  of  the  turns  S,  the  total 
ampere-wires  in  the  cross-section  d  X  I  is  (*S/).  For  a  first  ap- 
proximation of  the  area  required,  it  is  well  to  assume  a  certain 
current  density  in  the  windings,  which  is  not  likely  to  cause  an 
excessive  heat  loss  and  therefore  an  unsafe  rise  of  temperature. 
The  following  figures  may  be  used : 

For  large  magnets  try  A  =  700,  or  (M)  =  1,800 

For  medium-sized  magnets  try  A  =  900,  or  (M)  =  1,400 

For  small  magnets  try  A  =  1,100,  or  (M)  =  1,1 50] 

where  A  =  current  density  in  amperes  per  square  inch,  and  (M) 
=  number  of  circular  mils  per  ampere.  The  relation  between 
these  quantities,  as  previously  explained  (Art.  9)  is, 

A  X  (M)  =  1.273  X  106 

By  assuming  the  current  density,  it  is  then  easy  to  calculate  the 
probable  cross-section  of  the  copper  in  the  coil.  This,  however, 
is  not  equal  to  the  product  d  X  I  because  the  winding  space  factor 


42  PRINCIPLES  OF  ELECTRICAL  DESIGN 

must  be  taken  into  account.  A  little  practice  will  enable  the 
designer  to  form  a  rough  idea  as  to  the  size  of  wire  that  will 
be  required,  this  being  of  small  diameter  for  high  voltages  and  of 
large  diameter  for  low  voltages.  He  can  select  a  probable  value 
of  the  space  factor  from  the  curves  of  Fig.  15.  The  cross-section 
of  the  coil  can  now  be  calculated  because, 

O  T 

JXd  =  A3<tf  (25) 

Any  convenient  relation  between  I  and  d  may  be  chosen,  but 
the  value  of  one  of  these  dimensions  is  usually  decided  upon  in 
the  first  instance.  It  is  well  to  avoid  making  the  depth  of  wind- 
ing, d,  more  than  3  in.,  even  in  large  magnets,  because  the  internal 
temperature  is  then  liable  to  become  excessive. 

The  size  of  the  wire  will  depend  upon  the  length  of  the  mean 
turn;  and,  with  a  known  value  for  d,  and  a  core  of  circular  cross- 
section,  we  have: 

Mean  length  per  turn  =  ir(D  +  d)  in.     (See  Fig.  16.) 

Applying  formula  (21)  for  the  resistance  of  a  copper  wire  at  a 
temperature  of  about  60°C.,  we  may  write, 

.  ,  length  in  inches       E 

resistance  =  -       —  -,  —  x—       -  =  y 

(ra)  / 

whence 

ir(D  +  d)S       E 
(m)          =  I 
and 


(m)  =  * 

til 

In  this  manner  the  size  of  the  wire  can  be  determined.  It 
should  be  noted  that,  for  a  given  excitation,  its  cross-section 
depends  only  upon  the  applied  potential  difference  and  the 
average  length  per  turn;  it  is  quite  independent  of  the  number 
of  turns  of  wire,  S.  That  this  must  necessarily  be  the  case  is 
seen  when  it  is  realized  that  for  every  increase  in  S,  the  resist- 
ance increases  in  like  manner,  causing  the  current  I  to  decrease 
by  a  proportional  amount. 

By  referring  to  a  wire  table  such  as  those  on  pages  34  and  35, 
the  standard  gage  size  nearest  to  the  calculated  cross-section 
can  be  chosen.  If  it  does  not  seem  close  enough  to  the  required 
size  for  practical  purposes,  the  coil  can  be  wound  with  two  sizes 


THE  MAGNETIC  CIRCUIT—  ELECTROMAGNETS    43 

of  standard  wire,  as  will  be  explained  shortly,  but  it  is  generally 
possible  to  modify  the  average  length  per  turn  and  so  obtain  the 
desired  result.  The  formula  (26)  can  be  written, 

(»  +  d}  -  _^L 

"  irXSI 
whence 

d-     E(m)        D 

-*xsr 

Now,  if  (A)  is  the  area  in  circular  mils  of  the  standard  size  of 
wire  it  is  proposed  to  use  —  instead  of  the  previously  calculated 
cross-section  (m)  —  the  required  ampere-turns  can  be  obtained 
by  making  the  depth  of  winding, 


*  -     -  » 

Now  estimate  (by  using  the  space  factor  curves,  or  by  calculation) 
the  number  of  turns  required  to  fill  the  spool  to  the  required 
depth,  and  calculate  the  total  resistance,  R,  and  the  current, 


A  convenient  rule,  which  usually  provides  sufficient  winding 
space  to  prevent  excessive  temperature  rise,  is  to  allow  1  sq. 
in.  of  winding  space  cross-section  for  every  500  ampere-turns 
required  on  the  coil.  This  simply  means  that  the  product 
A  X  sf  of  formula  (25)  is  taken  as  500. 

Winding  Shunt  Coils  With  Two  Sizes  of  Wire.—  For  a  definite 
mean  length  per  turn,  the  exact  ampere-turns  required  on  a 
magnet  can  always  be  obtained  with  standard  sizes  of  wire  by 

N  —     —3  Feet  of  A  Ohms—       —  *+*-y  Feet  of  B  Ohms  --  H 

i  -  !  -  1 


One  Foot  —  R    Ohms 


FIG.  17. — Two  sizes  of  wire  in  series. 

using,  if  necessary,  two  wires  of  different  diameter  connected  in 
series  or  in  parallel.  The  series  connection  is  most  usual  for 
magnets  or  field  coils  to  be  connected  across  a  definite  voltage. 
Let  R  stand  for  the  ohms  per  foot  length  of  wire  to  give  the 
required  excitation  at  the  proper  temperature;  A  =  ohms  per 
foot,  at  the  same  temperature,  of  the  standard  wire  of  larger  size; 


44  PRINCIPLES  OF  ELECTRICAL  DESIGN 

and  B  =  the  corresponding  resistance  of  the  smaller  standard 
size  of  wire.  It  is  proposed  to  make  up  the  resistance  R  by 
connecting  x  feet  of  A  ohms  per  foot  in  series  with  y  feet  of  B 
ohms  per  foot,  as  indicated  in  Fig.  17.  Thus, 

xA  +  yB  =  R 
also 

x  +  y  =  1  or,  y  =  I  -  x 

whence 

xA  +  B  -  xB  =  R 
and 

.  =  (28) 


If  it  is  preferred  to  work  with  cross-sections  in  circular  mils, 
instead  of  resistances  in  ohms  per  foot,  we  can  put  the  relation  of 
formula  (28)  in  the  form 


(A)       (B) 
_  (A)       (B)  -  (m) 
~  (m)  X  (B)  -  (A) 
where   (m)  =  calculated  circular  mils, 

(A)  =  circular  mils  of  larger  standard  wire, 

(B)  =  circular  mils  of  smaller  standard  wire. 

When  winding  with  two  sizes  of  wire  in  series,  it  is  usual  to  put 
the  smaller  wire  on  the  outside  where  the  heat  will  be  most  readily 
dissipated. 

11.  Heat  Dissipation  —  Temperature  Rise.  —  The  winding,  if 
calculated  as  explained  in  the  preceding  article,  will  furnish  the 
required  excitation;  but  it  is  possible  that  the  estimated  value  for 
the  current  density  may  result  in  a  temperature  so  high  as  to 
injure  the  insulation,  or  so  low  as  to  render  the  cost  of  the  magnet 
—  owing  to  excess  of  copper  —  commercially  prohibitive.  The 
highest  temperature  will  be  attained  somewhere  inside  the  coil, 
and  it  is  not  easily  calculated;  the  temperature  as  measured  by  a 
thermometer  on  the  outside  of  the  coil  is  only  a  rough  guide  to 
that  of  the  hottest  part.  The  average  temperature  is  also  higher 
than  the  outside  temperature;  it  can  be  ascertained  by  meas- 
uring the  resistance  of  the  coil  hot  and  cold.  The  maximum 
temperature  can  be  measured  only  by  burying  thermometers  or 


THE  MAGNETIC  CIRCUIT—  ELECTROMAGNETS    45 

test  resistances  in  the  center  of  the  coil  when  it  is  being  wound. 
The  depth  of  winding  has  much  to  do  with  the  relation  between 
outside  and  inside  temperatures.  This  depth  should  rarely 
exceed  3  in.,  and  a  long  coil  of  small  thickness  will,  obviously, 
have  a  much  more  uniform  temperature  than  a  short  thick  coil 
of  the  same  number  of  turns. 

As  a  rough  indication  of  what  may  be  expected  in  the  matter 
of  internal  temperatures,  it  may  be  stated  that,  in  magnet  coils 
of  average  size,  the  mean  temperature  might  be  1.4  times,  and 
the  maximum  temperature  1.65  times,  the  external  temperature. 
The  maximum  allowable  safe  temperature  for  cotton-covered 
wires  is  95°C.,  and  as  this  may  be  reached  when  the  outside  tem- 
perature is  40°C.  above  that  of  the  surrounding  medium,  a 
maximum  rise  of  temperature  of  40°  or  45°C.,  as  measured  at 
the  hottest  accessible  part  of  the  finished  coil,  is  usually  specified. 
If  the  calculated  temperature  rise  is  in  excess  of  this,  the  coil 
must  be  re-designed  in  order  to  increase  the  cooling  surface  or 
reduce  the  PR  loss. 

The  calculation  of  temperature  rise  is  based  largely  upon 
coefficients  which  are  the  result  of  tests,  preferably  conducted  on 
coils  of  the  same  type  and  size  as  the  one  considered.  The  cool- 
ing surface  of  a  magnet  winding  of  the  type  shown  in  Fig.  16,  page 
41,  may  be  taken  as  the  outside  cylindrical  surface  only;  or  this 
outside  surface  plus  the  area  of  the  two  ends;  or,  again,  the  whole 
surface,  not  omitting  the  inside  portion  in  proximity  to  the  iron 
of  the  magnet  core.  This  is  largely  a  matter  of  individual  choice 
based  on  experience  gained  with  similar  types  of  coil,  and  the 
heating  coefficient  will  necessarily  have  a  different  value  in 
each  case. 

E2 
The  watts  lost  amount  to  PR,  or  El,  or  p  .     The  heating 

coefficient  is  the  cooling  surface  necessary  to  dissipate  one  watt 
per  degree  difference  of  temperature  between  the  outside  of  the 
winding  and  the  surrounding  air.  Thus 

k-^ 
~  PR 

and 


(30) 

where  T  is  the  temperature  rise  in  degrees  Centigrade;  fcis  the 
heating  coefficient,  which  can,  if  preferred,  be  properly  defined 


46  PRINCIPLES  OF  ELECTRICAL  DESIGN 

as  the  degrees  Centigrade  rise  in  temperature  when  the  loss  in 
watts  is  equal  to  the  cooling  surface  in  square  inches;  and  A  is 
the  actual  cooling  surface  expressed  in  square  inches.  The  area 
of  this  cooling  surface  will  be  reckoned  as  the  sum  of  the  outside 
and  inside  perimeters  multiplied  by  the  length  of  the  coil,  plus 
the  area  of  both  ends  of  the  coil.  The  temperature  rise  is 
found  to  differ  very  little  whether  the  coil  is  surrounded  entirely 
by  air,  or  provided  with  an  iron  core,  and  for  this  reason  the 
writer  prefers  to  consider  the  total  external  area  of  the  coil  as 
the  cooling  surface.1 

.  The  heating  coefficient  k  is  not  a  constant,  even  for  a  given  size 
and  shape  of  magnet.  It  is  a  function  of  the  difference  of 
temperature  between  the  coil  surface  and  the  surrounding 
medium;  it  also  depends  upon  the  material  of  the  spools  or 
bobbins,  on  the  insulating  varnish  and  wrappings  (if  any),  and 
other  details  of  construction.  Assuming  a  surface  temperature 
rise  of  about  40°C.  and  open  type  coils — that  is  to  say,  coils 
with  ends  and  outside  surface  exposed  to  the  air — finished  with 
a  coat  or  two  of  varnish  over  the  cotton-covered  wire,  the  coeffi- 
cient k  might  lie  between  160  and  200,  with  an  average  value  of 
180.  With  a  temperature  rise  of  only  20°C.  the  average  value 
of  k  should  be  taken  as  190. 

In  the  case  of  iron-clad  coils  such  as  those  found  in  many 
designs  of  lifting  magnets  and  magnetic  clutches,  the  final 
internal  temperature  will  depend  largely  on  the  shape  and  thick- 
ness of  the  surrounding  iron,  and  on  the  total  radiating  surface; 
but,  for  approximate  calculations,  the  same  coefficient  may  be 
used  as  for  the  open  coils,  bearing  in  mind  that,  in  all  cases, 
the  temperature  rise  T  of  formula  (30)  is  that  of  the  outside  layer 
of  wire,  and  the  area  A  is  that  of  the  total  external  surface  of  the 
copper  coil. 

12.  Intermittent  Heating. — Without  attempting  to  discuss 
exhaustively  the  effects  of  intermittent  service,  the  two  extreme 
cases  may  be  considered :  (a)  the  apparatus  is  alternately  carrying 
the  full  current,  and  carrying  no  current,  during  short  periods  of 
time  extending  over  many  hours,  so  that  the  total  cooling  surface 
is  the  factor  of  importance;  and  (6),  the  apparatus  is  in  use  at 

1  This  is  the  recommendation  of  MR.  G.  A.  LISTER  in  his  excellent  paper 
published  in  the  British  Journal,  Inst.  E.  E.,  vol.  38,  p.  402,  to  which  the 
reader  is  referred  if  he  wishes  to  pursue  further  the  subject  of  *magnet- 
coil  heating. 


THE  MAGNETIC  CIRCUIT— ELECTROMAGNETS    47 

only  rare  intervals  of  time,  with  long  periods  allowed  for  cooling, 
so  that  the  factor  of  importance  is  the  capacity  for  heat. 

Case  (a). — During  a  period  of  1  hr.,  the  current  is  passing 
through  the  magnet  coil  for  a  known  short  interval  of  time, 
and  is  then  switched  off  for  another  known  period,  so  that  out  of 
a  total  of  60  min.,  the  current  flows  through  the  coil  during  h 
mm.  only;  the  temperature  rise  can  then  be  caculated,  as 
previously  explained,  by  making  the  assumption  that  the  watts 

to  be  dissipated  are  not  W  =  PR;  but  Wh  =  -  ~™ 

This  method  cannot  safely  be  used  if  the  "on"  and  "off" 
periods  are  long;  but  no  general  rule  can  be  formulated  in  this 
connection  because  the  size  of  the  magnet  is  an  important 
factor. 

Case  (b). — If  used  only  at  rare  intervals  of  time,  with  long 
periods  allowed  for  cooling  down,  a  magnet  coil  can  be  worked 
at  very  high  current  densities.  The  temperature  rise  is  then 
determined  solely  by  the  specific  heat  of  the  copper,  and  its  total 
weight  or  volume. 

The  specific  heat  of  a  substance  is  the  number  of  calories  re- 
quired to  raise  the  temperature  of  1  gram,  1°C.  The  specific 
heat  of  water  at  ordinary  temperatures  being  taken  as  unity, 
that  of  copper  is  about  0.09.  One  calorie  will  raise  1  gram  of 
water  1°C.;  and  since  1  calorie  is  equivalent  to  42  X  108  ergs 
(or  dyne-centimeters),  it  follows  that,  to  raise  1  gram  of  copper 
1°C.  in  1  sec.,  work  must  be  done  at  the  rate  of  0.09  X  42  X  10* 
ergs  per  second.  But  1  watt  is  the  rate  of  doing  work  equal  to 
107  ergs  per  second;  and  1  Ib.  =  453.6  grams;  this  leads  to  the 
conclusion  that  the  power  to  be  expended  to  raise  1  Ib.  of  copper 
1°C.  in  1  sec.  is 

0.09  X  42  X  106  X  453.6 

— T7y7 —  -  =  171.5  watts. 

A  cubic  inch  of  copper  weighs  0.32  Ib.,  and  (173  X  0.32)  or 
55  watts  will  therefore  raise  the  temperature  of  1  cu.  in.  of  copper 
1°C.  in  1  sec. — assuming  no  heat  to  be  radiated  or  conducted 
away  from  the  surface  of  the  coil. 

In  this  manner  it  is  possible  to  calculate  how  long  an  electro- 
magnet for  occasional  use  can  be  left  in  circuit  without  damage 
to  insulation.  A  temperature  rise  of  50°  to  55°C.  is  generally 
permissible  in  making  calculations  on  the  heat-capacity  basis. 


CHAPTER  III 
THE  DESIGN  OF  ELECTROMAGNETS 

13.  Introductory. — The  object  of  this  chapter  is  partly  to 
summarize  and  coordinate  what  has  already  been  discussed; 
but  mainly  to  familiarize  the  reader  with  the  laws  of  the  magnetic 
circuit  and  the  simple  computations  which  will  enable  him  to 
proportion  the  iron  cores  and  calculate  the  field  windings  of 
electric  generators.  A  little  practice  in  the  design  of  the  simple 
forms  of  lifting  magnet,  or  magnetic  brake,  will  be  of  the  greatest 
value  in  illustrating  the  practical  application  of  the  fundamental 
principles  underlying  the  design  of  all  electromagnetic  machinery. 
The  designer  who  wishes  to  specialize  in  lifting  magnets,  magnetic 
clutches,  and  electromagnetic  mechanisms  generally,  must  pursue 
his  studies  elsewhere :  he  is  referred  to  other  sources  of  information 
such  as  MR.  C.  R.  UNDERBILL'S  book  on  electromagnets.1 
There  are  many  matters  of  interest,  such  as  the  means  of  ob- 
taining quick,  or  slow,  action  in  magnets;  equalizing  the  pull 
over  long  distances;  special  features  of  alternating-current  electro- 
magnets; and  the  mechanical  devices  introduced  to  attain  specific 
ends,  but  none  of  these  can  receive  adequate  attention  here. 

In  the  design  of  electromagnets  with  movable  armatures  or 
plungers,  the  work  to  be  done  is  usually  reckoned  as  the  initial 
or  starting  pull,  in  pounds,  multiplied  by  the  travel,  in  inches. 
Many  designs,  of  varying  sizes  and  costs,  can  be  made  to  comply 
with  the  terms  of  a  given  specification,  and  the  main  object 
of  the  designer  should  be  to  put  forward  the  design  of  lowest 
cost  which  will  fulfil  the  conditions  satisfactorily.  It  is  not 
proposed  to  devote  much  space,  either  here  or  elsewhere,  to  the 
detailed  discussion  of  the  commercial  aspects  of  design;  but  it 
is  well  to  emphasize  the  fact  that  a  designer  who  does  not  con- 
stantly bear  in  mind  the  factors  of  first  cost  and  cost  of  upkeep, 
is  of  little  or  no  value  to  the  manufacturer.  In  the  design  of 
electromagnets,  especially  of  the  larger  sizes,  the  material 

1  "Solenoids,  Electromagnets  and  Electromagnetic  Windings:"  D.  VAN 
NOSTRAND  Co. 

48 


THE  DESIGN  OF  ELECTROMAGNETS 


49 


cost  is  the  main  item,  and  the  total  cost  of  iron  and  copper  is 
a  good  guide  to  the  cost  of  the  finished  magnet,  when  it  is 
desired  merely  to  compare  alternative  designs  based  on  a  given 
specification. 

It  is  an  easy  matter  to  estimate  the  volume  and  weight  of 
materials  in  so  simple  a  design  as  a  lifting  magnet,  and  although 
formulas  can  be  developed  which  aim  to  give  the  proportions 
and  sizes  for  the  most  economical  design,  these  are  usually  of 
doubtful  value,  and  it  is  generally  simpler  to  apply  a  little 
common  sense  and  the  engineering  judgment  which  will  come 
with  practice,  and  try  two  or  three  designs  with  different  pro- 
portions before  selecting  the  one  that  seems  most  suitable  in  all 
respects,  not  omitting  the  important  item  of  initial  cost. 

14.  Short-stroke  Tractive  Magnet. — With  a  design  of  plunger 
magnet  as  shown  in  Fig.  18,  there  is  not  much  magnetic  leakage, 


Brass  Ring 


FIG.  18. — Plunger  or  iron  clad  magnet. 

because  the  travel  of  the  plunger,  or  length  of  air  gap,  is  small  in 
comparison  with  the  area  of  the  pole  faces.  Given  a  definite  total 
amount  of  flux  to  produce  the  required  pull,  the  cross-section  of 
the  various  parts  of  the  magnetic  circuit  is  readily  calculated. 


50  PRINCIPLES  OF  ELECTRICAL  DESIGN 

Various  proportions  can  be  tried,  also  different  values  of  the 
magnetic  density  in  the  air  gap.  The  pull  per  square  inch  de- 
pends upon  B2;  but,  by  forcing  the  density  up  to  high  values,  the 
ampere-turns  required  become  excessive,  and  the  weight  and  cost 
of  the  copper  coils,  prohibitive.  More  will  be  learned  by  trying 
various  proportions  and  roughly  estimating  the  cost,  than  by  a 
lengthy  discussion  of  the  manner  in-which  the  various  dimensions 
are  dependent  upon  each  other.  It  will  probably  be  found  that 
the  most  economical  initial  density  will  not  exceed  11,000 
gausses;  and  (by  formula  16,  Art.  8)  the  pull,  in  pounds  per 
square  inch,  is 

(11, OOP)2  _  7n  „ 
1,730,000  ~ 

thus,  with  the  usual  cylindrical  core, 

ird2 
total  force,  in  pounds  =  F  —  70  —- 

whence 

d  =  0.135  \/F  (31) 

The  magnet  can  now  be  sketched  approximately  to  scale,  and  the 
necessary  ampere-turns  computed,  all- as  previously  explained  in 
Art.  4.  Although  Fig.  18  shows  a  very  short  air  gap,  the  same 
methods  apply  to  the  calculation  of  magnets  with  longer  air  gap, 
provided  this  is  not  so  great  as  to  cause  excessive  magnetic 
leakage.  A  practical  rule  which  determines  the  minimum  length 
of  the  winding  space  is  that  this  length,  h,  should  never  be  less 
than  twice  the  air-gap  length,  I. 

15.  Magnetic  Clutch. — The  design  of  a  magnetic  clutch  to 
transmit  power  between  a  shaft  and  pulley  or  any  piece  of  rotat- 
ing machinery,  is  generally  similar  to  that  of  the  circular  type 
of  lifting  magnet.  Fig.  19  shows  a  common  type  of  magnetic 
clutch  with  conical  bearing  surfaces,  although  the  conical  shape 
is  not  essential,  and  the  wedge  action  of  the  cone-shaped  rings  is 
not  relied  upon  to  increase  the  pressure  between  the  surfaces  in 
contact.  When  the  two  iron  surfaces  are  held  together  by  the 
action  of  the  exciting  coil,  the  flux  density  over  the  area  between 
the  two  annular  pole  faces  must  be  such  as  to  produce  a  force 
that  will  prevent  slipping  between  these  faces.  A  factor  of 
safety  of  2.5  to  3  is  generally  allowed. 


THE  DESIGN  OF  ELECTROMAGNETS 


51 


Let  R  =  mean  radius,  in  feet. 

A  =  area  of  all  North,  or  of  all  South,  polar  surfaces  in 

contact  (square  inches). 

P  =  pressure  in  pounds  per  square  inch  of  contact  surface. 
N  =  revolutions  per  minute. 

c  =  coefficient  of  friction,  the  meaning  of  which  is  that 
c  X  PA  is  the  tangential  force  which  will  just  produce 
slipping. 
then 

,  PAX  cX  2irR  XN 

hp'  =  -33,000 


FIG.   19. — Magnetic  clutch. 

which  gives  the  horsepower  that  the  clutch  will  transmit  just 
before  slipping  occurs.  If  k  is  the  safety  factor,  and  (hp.)  is 
the  horsepower  which  has  to  be  transmitted,  the  value  of  hp. 
to  insert  in  formula  (32)  should  be  (hp.)  X  k.  Note  also  that 

B2 
P  =  T-s«>r          and  different  values  of  B  can  be  tried;   these 


52 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


values  should  be  fairly  high,  but  they  will  depend  upon  whether 
the  magnetic  circuit  is  of  cast  iron  or  steel.  The  unknown 
quantities  are  then  R  and  A,  and  equation  (32)  can  be  put  in  the 
form 

33,000  X  (hp.)  X  k  X  1,730,000 


R  X  A  = 


2wNCB2 

91  X  108  X  (hp.)  X  k 
NCB2 


(33) 


where  B  is  in  gausses,  and  the  coefficient  of  friction,  c  (for  dry 
surfaces),  may  be  taken  from  the  accompanying  table.  If  the 
surfaces  are  lubricated  with  oil  or  grease,  the  friction  coefficient 
may  be  lowered  as  much  as  40  per  cent.  One  reason  for  using  a 
large  factor  of  safety  is  to  allow  for  the  possibility  of  dirt  or  oil 
getting  between  the  surfaces  in  contact. 

TABLE  GIVING  APPROXIMATE  VALUES  OF  THE  COEFFICIENT  OF  FRICTION,  c 


Pressure,  Ib. 
per  sq.  in. 

Wrought  iron  on 
wrought  iron 

Wrought  iron  on 
cast  iron 

Cast  iron  on  steel 

Cast  iron  on  cast 
iron 

50 

0.151 

0.182 

0.188 

0.113 

100 

0.187 

0.222 

0.237 

0.140 

150 

0.217 

0.254 

0.275 

0.165 

200 

0.259 

0.280 

0.304 

0.190 

The  formula  (33)  permits  of  either  quantity  R  or  A  being 
calculated  when  one  of  them  is  known  or  assumed.  It  is  the 
business  of  the  designer  to  determine — usually  by  trial — the 
dimensions  which  will  give  the  best  results.  So  far  as  cost  is 
concerned,  a  large  diameter  may  show  a  saving  in  materials;  but 
the  labor  cost — not  omitting  the  cost  of  patterns  when  but  few 
castings  are  required — should  also  be  considered. 

At  times  when  slipping  occurs,  as  in  a  magnetic  brake,  or 
when  throwing  in  a  magnetic  clutch  while  there  is  some  relative 
movement  between  the  two  parts,  there  is  a  powerful  retarding, 
or  driving,  action  as  the  case  may  be,  due  not  to  the  direct 
magnetic  pull  between  the  surfaces  in  contact,  but  to  the  fact  that 
eddy  currents  are  produced  in  the  polar  faces  on  account  of 
the  cutting  of  the  magnetic  lines.  This  cutting  of  flux  is  similar 
to  what  occurs  in  the  unipolar,  or  so-called  homopolar,  type  of 
D.C.  generator,  where  the  currents  are  confined  to  certain  paths 
and  collected  by  means  of  sliding  brushes. 


THE  DESIGN  OF  ELECTROMAGNETS 
\ 

NUMERICAL  EXAMPLES 


53 


16.  Horseshoe  Lifting  Magnet. — Assume  the  specified  condi- 
tions to  be  as  follows: 

Initial  pull  =  200  Ib. 

Travel  of  armature  (being  the  length  of  the  single  air  gap)  = 
0.35  in. 

B.C.  voltage  =  110. 

Allowable  temperature  rise  =  40°C. 


K---K;— -H 


Pole  Face  2  Square 

FIG.  20. — Horseshoe  magnet. 

The  temperature  to  be  taken  on  the  outside  surface  of  the 
exciting  coils,  after  a  sufficient  time  has  elapsed  for  the  final  tem- 
perature to  be  reached. 

The  required  magnet  might  be  generally  as  shown  in  Fig.  20, 
where  the  iron  limbs  are  of  circular  section  with  square  pole  pieces. 
These  limbs  may  be  steel  castings,  or  they  can  be  turned  down 
from  square  bar  iron.  Cast  iron  would  not  be  a  suitable  material, 


54  PRINCIPLES  OF  ELECTRICAL  DESIGN 

because  the  large  cross-section  necessary  to  keep  the  flux  density 
within  reasonable  limits  would  lead  to  an  unnecessary  and 
wasteful  increase  in  the  weight  of  the  copper  coils. 

Applying  formula  (16)  of  Art.  8,  page  29,  the  pull,  expressed 
in  terms  of  the  air-gap  density  is 

2AB* 
~  1,730,000 

where  B  is  in  gausses,  and  A  is  the  area  of  one  polar  face,  expressed 
in  square  inches.  Thus, 

200         ***** 

~  1,730,000 
whence, 

13,150 
«"-.-]&-  (34) 

This  relation  between  size  of  pole  face  and  the  air-gap  density 
must  exist  if  the  pull  of  200  Ib.  is  to  be  obtained,  but  the  density  B 
can  be  varied  within  wide  limits.  It  is  obvious  that  high  values 
of  B  are  advantageous  in  so  far  as  they  reduce  the  weight  and 
cost  of  the  iron  in  the  magnet;  but  since  the  initial  air-gap  length 
remains  constant,  the  necessary  ampere-turns  will  increase 
almost  in  direct  proportion  to  any  increase  of  B.  The  economical 
value  of  the  flux  density,  B,  cannot  be  immediately  determined; 
and  although  formulas  for  minimum  cost  can  be  developed,  they 
become  unwieldy  and  unpractical  when  all  the  important 
factors  are  taken  into  account.  On  the  other  hand,  if  all  deter- 
mining factors — including  such  items  as  cooling  surfaces  and 
magnetic  leakage — are  not  taken  into  consideration,  the  formulas 
are  very  inaccurate  and  not  of  general  application.  It  is  very 
interesting  to  develop  approximate  formulas  for  use  in  arriving  at 
the  economical  dimensions  of  any  particular  type  of  electro- 
magnetic apparatus,  and  the  reader  may  learn  much  by  trying 
to  put  the  various,  and  frequently  conflicting,  requirements  in  the 
form  of  a  mathematical  equation;  but  we  shall  follow  here  the 
method  adopted  by  a  large  number  of  experienced  designers, 
which  consists  in  trying  what  seems  a  probably  value  for  one  of 
the  unknowns,  and  then  checking  results  by  assuming  a  larger 
and  a  smaller  value  for  the  unknown  quantity. 

Since  very  low  values  of  B  will  lead  to  great  weight  of  iron, 
and  very  high  values  will  lead  to  an  increased  weight  of  copper, 
it  is  safe  to  assume  that  B  will  not  be  less  than  5,000  or  more  than 


THE  DESIGN  OF  ELECTROMAGNETS  55 

10,000  gausses.  We  shall  select  the  value  of  7,000  for  a  trial 
design,  and  carry  this  through  to  completion,  although  it  would 
usually  be  preferable  to  carry  at  least  three  designs  through  to 
the  point  where  it  becomes  clear  that  one  of  these  is  distinctly 
preferable  to  the  others  from  the  point  of  view  of  economy;  this 
being  the  main  consideration  which  the  designer  must  always  bear 
in  mind. 

Putting  for  B,  in  formula  (34),  the  value  7,000  gausses,  the  side 
w  of  the  (square)  pole  shoe  is  found  to  be  1.88  in.  Let  us  make 

13  150 
this  2  in.;  whence  B  =  —£—  =  6,570.     The  thickness  12  of  the 

pole  shoe  must  be  small  in  order  to  keep  down  the  magnetic 
leakage;  a  value  of  Y±  in.  for  12  should  be  satisfactory. 

The  diameter,  d,  of  the  magnet  cores  under  the  winding  is 
obtained  by  assuming  a  leakage  factor  and  a  suitable  flux  density 
in  the  iron.  The  leakage  factor  (refer  Art.  7,  page  28,  for  defi- 
nition) might  be  about  1.5  in  a  magnet  of  this  type  and  size;  and 
the  density  in  the  core  of  magnet  steel  or  wrought  iron,  may  be 
as  high  as  90,000  lines  per  square  inch.  Thus; 

TT  1.5X4X  6,570  X  6.45 

4  90,000 

whence  d  =  1.9  or  (say)  d  =  1.875  in. 

Summing  up  the  quantities  so  far  determined,  we  have: 
w  =  2  in. 

h  =  K  in. 
d  =  1%  in. 

B  (air-gap  density)  =  6,750  gausses. 

4>  (useful  or  effective  flux  per  pole)  =  6,750  X  6.45  X  2  X  2 

=  174,000  maxwells. 

Depth  of  Winding. — The  length  I  of  the  winding  space,  and  the 
thickness  of  winding,  t,  will  depend  upon  the  ampere-turns  neces- 
sary to  produce  the  desired  flux  density  in  the  air-gap,  and  also 
upon  the  allowable  current  density,  A,  in  the  copper  of  the 
exciting  coils.  The  relation  of  I  to  t  is  not  determined  by  the 
amount  of  the  ampere-turns,  since  this  only  calls  for  a  sufficient 
cross-section,  or  product  I  X  t.  The  thickness  t  should  not  exceed 
3  in.,  because  a  greater  thickness  may  lead  to  excessive  tem- 
peratures inside  the  coil ;  but  the  most  suitable  dimensions  of  the 
coil  are  really  determined  by  the  current  density,  the  winding 
space  factor,  and  the  cooling  surface  necessary  to  prevent 


56  PRINCIPLES  OF  ELECTRICAL  DESIGN 

excessive  temperature  rise.  It  is  usual  to  assume  a  value  for  the 
thickness,  t,  which  may  be  something  more  than  one-third  of  the 
diameter  of  the  core,  with  the  previously  mentioned  limit  of 
about  3  in.  Thus,  even  if  d  were  greater  than  9  in.,  the  depth  of 
winding  should,  preferably,  not  exceed  3  in.  In  the  present 
case  t  should  be  about  1.875  -5-  3  =  0.625  in.  Let  us  try  t  = 
%in. 

Current  Density  in  Windings.  —  If  a  suitable  value  for  the 
current  density  in  the  windings  can  be  chosen,  it  will  be  an  easy 
matter  to  determine  the  length,  I,  of  the  winding  space,  and  so 
complete  the  preliminary  design. 

Let  A  =  the  current  density  (amperes  per  square  inch)  . 
R"  =  the  resistance,  in  ohms,  between  opposite  faces  of  an 
inch  cube  of  copper.     By  formula   (29)   of  Art.   9, 

*"  -        at  60°c- 


sf  =  the  winding  space  factor,  as  given  in  Fig.  15,  page  40. 

As  the  size  of  wire  is  not  yet  known,  a  probable  value  of 

0.5  will  be  chosen  for  this  factor,  in  the  preliminary 

calculations. 
T  =  the  allowable  temperature  rise,  being  40°C.  in   this 

example. 
k  =  the  cooling  coefficient,  being  denned  in  Art.  9  formula 

cooling  surface 

(30),  as  T  X  u     i.    •      —  -  ?•    An  average  value 

watts  to  be  dissipated 

of  180  may  be  taken  for  k. 

Equating  the  I2R  losses  with  the  watts  that  can  be  dissipated 
without  exceeding  the  temperature  limit,  we  can  write, 

T 
total  surface  of  coil  X  IT  =  watts  lost 

=  R"  A2  X  cubic  inches  of  copper 
or 


[2lr(d  +  0X2  +  4fcr(d  +  0]  =  #"A2  X  2Ur(d  +  t)  X  sf 

whence 

. 
~ 


In  order  to  eliminate  I,  we  may  consider  t  in  the  numerator  to 
be  negligible,  since,  in  this  particular  design,  with  an  air  gap  of 


THE  DESIGN  OF  ELECTROMAGNETS  57 

considerable  reluctance,  t  will  be  small  in  comparison  with  I. 
The  approximate  value  of  A  will  then  be, 


<36) 


2  X  40  X  1,273,000 
180X  1  X0.75  X  0.5 
=  (say)  1,250 

Length  of  Winding  Space.  —  The  ampere-turns  required  for  the 
double  air-gap  only,  i.e.,  not  including  those  required  to  overcome 
the  reluctance  of  the  iron  portions  of  the  magnetic  circuit,  will  be, 
by  formula  (5)  of  Art.  4, 

(SI)g  =  2.025  X  21, 

=  2.02  X  6,570  X  2  X  0.35 
=  9,300. 

The  ampere-turns  for  the  iron  part  of  the  magnetic  circuit  cannot 
be  calculated  accurately  until  the  length  /  and  the  actual  leakage 
factor  have  been  determined;  but,  since  the  air-gap,  in  this  case, 
offers  far  more  reluctance  than  the  remaining  portions  of  the 
magnetic  circuit,  we  shall  assume  the  iron  portions  to  require 
only  one-tenth  of  the  air-gap  ampere-turns.  Thus, 

total  SI  (both  spools)  ==  10,230  approx. 

We  are  now  able  to  solve  for  the  length  of  the  winding  space, 
which  is, 

SI  total 
=          X  «/' 

10,230 


2  X  0.75  X,250  X  0.5 

=  10.9  or  (say)  11  in. 

Before  proceeding  further  with  the  design,  it  will  be  well  to  see 
whether  the  long  magnet  limbs  will  not  be  the  cause  of  too  great  a 
leakage  flux.  If  the  leakage  factor  is  much  in  excess  of  the 
assumed  value  (1.5)  there  is  danger  of  saturating  the  cores  under 
the  windings,  and  so  limiting  the  useful  flux  available  for  drawing 
up  the  armature  Let  the  distance  between  the  windings  be  2  in., 
as  indicated  on  Fig.  20;  this  gives  all  necessary  dimensions  for 
calculating  the  permanence  of  the  leakage  paths. 

Calculation  of  Leakage  Flux.  —  The  total  leakage  flux  between 


58  PRINCIPLES  OF  ELECTRICAL  DESIGN 

the  two  magnet  limbs  should  be  considered  as  made-up  of  two 
parts  : 

(a)  The  flux  leakage  from  pole  shoe  to  pole  shoe,  which  is  due 
to  the  total  m.m.f.,  available  for  the  air  gap, 

(6)  The  flux  leakage  between  the  circular  cores  under  the 
windings,  which  increases  in  density  from  the  yoke  to  the  pole 
pieces  and  is  equal  to  the  average  m.m.f.  X  the  permeance  of  the 
air  paths  between  the  two  iron  cylinders.  This  average  m.m.f. 
will  be  approximately  one-half  the  total  m.m.f.  of  the  exciting 
coils. 

For  the  permeance  of  the  paths  comprised  under  (a),  we  have: 

1.  Between  opposing  rectangular  faces, 

6.45  X  2  X  0.25 
2.54  X  3.375 

2.  Between  the  two  pairs  of  faces  parallel  to  the  plane  of  the 
paper  (Fig.  20),  by  formula  (10)  Art.  5, 

A 


0.25  X  2.54  v  /TTX  2  +  3.375  \ 


7T 

=  0.425. 


X2.31oglo 


Neglecting  the  flux  lines  that  may  leak  out  from  the  pole  faces 
farthest  removed  from  each  other,  and  also  those  between  the 
small  ledges  caused  by  the  change  from  square  pole  piece  to 
circular  section  under  the  coil,  the  total  leakage  flux  between  pole 
pieces  is 

3>p=  m.m.f.  X  CPi  +  P2) 
=  0.47T  X  9,300  X  0.801 
=  9,350  maxwells. 

The  permeance  of  the  air  paths  comprised  under  (b)  may  be 
calculated  by  applying  formula  (13)  of  Art.  5.     Thus, 

T  X  11  X  2.54 


2.31og10f  h875 


=  51 


,3.5  +  1.875-  V(3^)2  +(2  X  3.5  X  1.875) / 

The  leakage  flux  between  magnet  cores  under  the  windings  is, 
^c=  m.m.f.   x^ 

0.47T  X  10,230  x 
o 

=  330,000  maxwells 


THE  DESIGN  OF  ELECTROMAGNETS  59 

and  the  total  leakage  flux  is 

$i  =  9,350  +  330,000  =  339,350  maxwells. 

The  leakage  factor  is 

170,000  +  339,350 


170,000 

which  is  greatly  in  excess  of  the  permissible  value,  unless  the 
cross-section  of  the  core  under  the  windings  is  increased  to  keep 
the  flux  density  within  reasonable  limits.  The  simplest  way  to 
reduce  the  amount  of  the  leakage  flux  is  to  shorten  the  magnet 
limbs,  and  although  the  long  limbs  with  no  great  depth  of  winding 
may  lead  to  economy  of  copper,  it  is  seen  to  be  necessary  in  this 
design  to  increase  the  depth  of  winding,  t,  in  order  to  reduce  the 
length,  I,  of  the  exciting  coils.  The  dimension  t  will  have  to  be 
more  than  doubled.  Let  us  make  this  1%  in.  and  at  the  same 
time  retain  the  full  section  of  2  in.  square  under  the  windings; 
that  is  to  say,  the  square  section  bar  will  be  carried  up  through  the 
coils  without  being  turned  down  to  a  smaller  section  as  in  the  trial 
design. 

Using  formula  (36)  l  to  calculate  the  current  density,  we  have, 


80  X  1,273,000 
:  \180X  1.75  X0.5 

=  (say)  800 
whence 

10,230 


2  X  1.75  X  800  X  0.5 
=  7.3  in. 

Let  us  try  I  =  7  in. 

Allowing  still  a  separation  of  2  in.  between  the  outside  surfaces 
of  the  windings,  the  distance  between  the  two  parallel  magnet 
cores  of  square  section  will  now  be  5.5.  in.  The  permeance 
between  the  opposite  faces  is 

7.25  X  2  X  6.45 
r\  -      — K~*;~v'~9~£A"~ 

and  between  the  sides  of  the  magnet  cores  (by  formula  10,  page 

25) 

7.25  X  2.54                       /TX2+5.5\_ 
" 2  =  2  X  -  -  X  2.6  logio  I  —   — £-£ )  —  o.y 

7T  \  O.O  / 

1  This  formula  and  also  the  correct  formula  (35)  are  applicable  to  rec- 
tangular as  well  as  to  circular  coils. 


60  PRINCIPLES  OF  ELECTRICAL  DESIGN 

The  total  leakage  flux  will  be  approximately 


=  100,000  maxwells. 

This  calculated  value  of  the  leakage  flux  should  be  slightly 
increased  because  the  total  permeance  between  the  two  magnet 
limbs  will  actually  be  greater  than  as  calculated  by  the  con- 
ventional formulas.  Let  us  assume  the  total  flux  to  be  125,000 
maxwells.  This  makes  the  value  of  the  leakage  factor. 

_  170,000  +  125,000  _ 
170,000 

The  maximum  value  of  the  density  in  the  iron  cores  will  be 

170,000  +  125,000 

—         —7—  -  =  73,600  lines  per  square  inch. 

Closer  Estimate  of  Exciting  Ampere  Turns. — The  modified 
magnet  will  now  be  generally  as  shown  in  Fig.  21.  The  ampere 
turns  required  to  overcome  the  reluctance  of  the  two  air  gaps 
have  already  been  calculated;  the  remaining  parts  of  the  mag- 
netic circuit  consist  of  the  two  magnet  limbs  under  the  windings, 
together  with  the  yoke  and  the  armature.  If  we  know  the 
amount  of  the  flux  through  the  iron  portions  of  the  circuit  we  can 
readily  calculate  the  flux  density,  and  then  ascertain  the  neces- 
sary m.m.f.  to  produce  this  density,  by  referring  to  the  B-H 
curves  of  the  material  used  in  the  magnet. 

In  the  magnet  cores  under  the  coils,  the  flux  density  varies 
from  a  minimum  value  near  the  poles  to  a  maximum  value  near 
the  yoke;  and  as  the  leakage  flux  is  not  uniformly  distributed 
over  the  length  lc  (Fig.  21),  it  would  not  be  correct  to  base  re- 
luctance calculations  upon  the  arithmetical  average  of  the  two 
extreme  densities,  even  if  the  flux  density  were  below  the  "knee" 
of  the  B-H  curve,  with  the  permeability,  /*,  approximately 
constant.  With  high  values  of  B}  the  length  of  the  magnetic 
core  should  be  divided  into  a  number  of  sections,  and  each  section 
treated  separately  in  calculating  the  required  ampere-turns. 
With  comparatively  low  densities,  as  in  this  example,  the 
calculation  can  be  made  on  the  assumption  of  an  average  density 
in  the  magnet  cores,  the  value  of  which  is 

D  V      '  p 

Dc   = o 


THE  DESIGN  OF  ELECTROMAGNETS 


61 


where  By  =  flux  density  at  end  near  yoke, 

and       Bp  =  flux  density  at  end  near  pole  pieces. 

The  total  ampere-turns  for  the  magnet  of  Fig.  21  can  now  be 
calculated  by  using  the  B-H  curve  of  Fig.  3  which  is  supposed  to 
apply  to  the  particular  quality  of  magnet  steel  or  iron  which  it 
is  proposed  to  use.  The  calculation  can  conveniently  be  put  in 
tabular  form  as  shown  below. 


TZ 


-0.36 


FIG.  21. — Horseshoe  magnet.     (Modified  design.) 


Part  of  circuit 

Length, 
inches 

Cross- 
section, 
square 
inch 

Total  flux, 
maxwells 

Density, 
lines  per 
square  inch 

SI  per 
inch 

SI, 
total 

Air  gaps  

0.7 

4.0 

170,000 

42,500 

9,300 

Armature  
Magnet  cores  
Yoke  

8.3 

14.5 
12  0 

2.5 
4.0 
4  0 

170,000 
220,000 
295,000 

68,000 
55,000 
73,600 

9.0 
7.5 
9  5 

75 
110 
115 

Total 

9,600 

Thus,  it  is  necessary  to  have  not  less  than  9,600  ampere-turns 
on  the  two  bobbins. 

Calculation  of  Windings. — The  formula  (26)  of  Art.  10  can  be 
written, 


(m)  = 


mean  length  of  turn  (inches)  X  SI 
E 


62  PRINCIPLES  OF  ELECTRICAL  DESIGN 

The  mean  length  of  turn  is  approximately  4(u>  +  £),  or  4  X 
3.75  =  15  in.;  and  the  potential  difference  across  the  two  coils 
is  110  volts;  thus, 

15  X  9,600 


Referring  to  the  wire  table  on  page  34,  the  wire  of  cross-section 
nearest  to  the  required  value  is  No.  18  B.  &  S.  gage,  because 
No.  19  will  be  too  small  to  provide  the  necessary  excitation. 
This  larger  wire  will  provide  a  factor  of  safety,  and  it  may  be  used 
if  the  watts  lost  and  the  temperature  rise  are  not  excessive. 

Calculation  of  Temperature  Rise.  —  The  space  factor  for  No. 
18  B.&  S.  D.C.C.  wire,  as  taken  off  the  curve  of  Fig.  15,  is  0.54. 
The  cross-section  of  the  copper  in  the  coil  is  therefore  7  X 
1.75  X  0.54  =  6.62,  and  the  number  of  turns  per  coil  will  be 
6.62/0.001276  =  5,180.  Other  required  values  are: 

Length  of  wire  in  one  coil  =  5,180  X  TS  =  6,500  ft. 

\.z 

Resistance  at  60°C.  =  6.5  X  7.42  =  48.2  ohms. 

Current  =  75-^  =1.14  amp. 

4o.Z 

Total  PR  loss  =  1.14  X  110  =  125  watts. 

Outer  surface  of  both  coils  =  2X7  X  4  X  5.5  =308 
Inner  surface  of  both  coils  =  2X7  X4X2  =112 
End  surfaces  of  both  coils  =  4  X  1.75  X  4  X  3.75  =  105 

Total  cooling  surface  ...................  ....  =  525  sq.  in. 

The  rise  of  temperature,  by  formula  (30)  Art.  11  taking 
k  =  180,  is 

1OK 

T  =  180  X         =  43°C. 


which  is  only  slightly  in  excess  of  the  specified  temperature  rise 
(40°C.).  Another  layer  or  two  of  winding  would  bring  the 
temperature  down  to  the  required  limit;  or,  if  preferred,  the 
length  of  the  coil  may  be  increased  by  a  small  amount  without 
appreciably  adding  to  the  reluctance  of  the  magnetic  circuit. 

It  should  be  mentioned  that  the  design  of  magnet  as  shown  in 
Fig.  21,  is  probably  larger  than  would  be  necessary  to  fulfil 
practical  requirements,  because  it  is  not  likely  that  the  full 
pressure  of  110  volts  would  be  maintained  across  the  terminals 


THE  DESIGN  OF  ELECTROMAGNETS  63 

for  many  hours.  The  magnet  would  be  designed  either  for  inter- 
mittent operation,  in  which  case  the  temperature  rise  might  be 
calculated  as  in  Case  (a)  of  Art.  12,  page  47,  or,  if  left  con- 
tinuously in  circuit,  a  resistance  would  automatically  be  thrown 
in  series  with  the  coil  windings  in  order  to  reduce  the  PR  loss 
and  effect  a  saving  of  copper  while  still  maintaining  the  required 
pull  of  200  Ib.  through  the  reduced  air  gap. 

Factor  of  Safety.  —  Seeing  that  the  coils  are  actually  wound  with 
a  wire  of  greater  cross-section  than  the  calculated  value,  the 
initial  pull  will  be  somewhat  greater  than  the  specified  200  Ib. 
The  actual  ampere-turns  are  5,180  X  2  X  1.14  ==  11,800,  and 
since  the  density  in  the  iron  is  not  carried  above  the  "knee" 
of  the  B-H  curve,  the  actual  flux  density  in  the  air  gap,  instead 

1  1  800 
of  being  6,570  gausses,  will  be  approximately  6,570  X  o 


8,070  or  (say)  8,000  gausses.     The  initial  pull  will  actually  be 

(Q  000")  2 
200  X  /6'570r2  =  30°  lb-  nearly.     This  factor  of  safety  of  1.5 

may  seem  excessive,  and  if  the  strictest  economy  of  material  is 
necessary,  the  coils  should  be  wound  with  a  wire  of  the  calcu- 
lated size,  or,  if  standard  gage  numbers  must  be  used,  as  would 
generally  be  the  case,  the  mean  length  of  turn  may  be  modified  by 
providing  a  greater  or  smaller  depth  of  winding  space.  As  an 
alternative,  two  sizes  of  wire  may  be  used  as  explained  in  Art.  10. 

Most  Economical  Design.  —  The  cost  of  materials  is  easily 
estimated  by  calculating  the  weight  of  iron  and  copper  sepa- 
rately. For  the  purpose  of  comparing  alternative  designs,  it 
is  usual  to  take  the  cost  of  copper  as  five  times  that  of  the  iron 
parts  of  the  magnet.  If  actual  costs  are  required,  the  figures 
would  be  about  20c.  per  pound  for  copper  wire,  and  4c.  per 
pound  for  the  magnet  iron. 

The  reader  will  recollect  that  this  design  has  been  worked 
through  on  the  assumption  that  about  6,500  gausses  would  be  a 
suitable  density  in  the  air  gap.  If  many  magnets  are  to  be  made 
to  the  one  design,  or  in  any  case  if  the  magnet  is  large  and  costly, 
the  designer  should  now  try  alternative  designs,  using  air-gap 
densities  of  (say)  4,000  and  8,000  gausses  respectively.  By  com- 
paring the  three  designs,  all  of  which  will  comply  with  the  terms 
of  the  specification,  he  will  be  able  to  select  the  one  which  can 
be  constructed  at  the  least  cost.  This  method  of  working  may 
seem  slow  and  tedious,  but  it  is  sure,  and  —  if  actually  tried  — 


64  PRINCIPLES  OF  ELECTRICAL  DESIGN 

will  be  found  to  involve  less  time  and  labor  than  might  be 
supposed.  The  student  following  the  courses  at  an  engineering 
college  does  not — unless  he  has  had  outside  experience — ap- 
preciate the  value  of  his  time.  Time  may  be  used,  abused,  or 
wasted;  and  when  a  concrete  and  definite  piece  of  work  has  to 
be  done,  the  time  spent  upon  it,  not  only  by  the  workman,  but 
also  by  the  designer,  may  be  of  no  less,  or  even  of  greater,  im- 
portance than  the  cost  of  the  materials.  The  case  in  point 
exemplifies  this.  If  the  required  magnets  are  -small,  and  but 
two  or  three  are  likely  to  be  wanted,  the  designer  should  not 
spend  much  time  on  refinements  of  calculation  and  in  endeavor- 
ing to  reduce  the  cost  of  manufacture  to  the  lowest  limit;  but  if 
the  magnets  are  of  large  size  and  several  hundred  will  be  re- 
quired, then  time  spent  by  the  designer  in  comparing  alternative 
designs  and  in  striving  to  reduce  material  and  labor  cost,  would 
be  amply  justified.  These  considerations  and  conclusions  may, 
to  many,  appear  elementary  and  obvious;  but  they  emphasize 
the  importance  of  what  is  generally  understood  by  "  engineering 
judgment"  which  is  rarely  acquired  or  rightly  valued  until  after 
the  student  has  left  school. 

Before  taking  up  the  design  of  another  form  of  magnet,  it 
may  be  well  to  state  that  the  method  of  procedure  here  followed 
in  the  case  of  a  horseshoe  magnet  is  not  put  forward  as  being 
necessarily  the  best,  or  such  as  would  generally  be  adopted  by 
an  experienced  designer.  It  serves  to  illustrate  much  that  has 
gone  before,  and  emphasizes  the  fact  that,  even  if  the  designer 
must  make  some  assumptions  and  do  a  certain  amount  of  guess- 
work at  the  beginning,  and  during  the  course,  of  his  design,  he 
can  always  check  his  results  when  the  work  is  completed,  and 
satisfy  himself  that  his  design  complies  with  all  the  terms  of  the 
specification. 

17.  Circular  Lifting  Magnet. — The  electromagnet  of  which 
Fig.  22  is  a  sectional  view  is  circular  in  form.  Its  function  is  to 
lift  a  ball  of  steel  weighing,  say,  4,000  lb.,  which,  on  the  opening 
of  the  electric  circuit,  will  fall  upon  a  heap  of  scrap  iron.  This 
device  is  referred  to  colloquially  as  a  "skull  cracker."  The 
diameter  of  a  solid  steel  sphere  weighing  4,000  lb.  is  approxi- 
mately 30  in.  If  the  outer  cylindrical  sheel — forming  one  of  the 
poles  of  the  magnet — has  an  average  diameter  of  21  in.,  it  will 
include  an  angle  of  90  degrees,  as  indicated  in  Fig.  22,  and  lead 
to  a  design  of  reasonable  dimensions.  If  the  required  width  of 


THE  DESIGN  OF  ELECTROMAGNETS 


65 


the  annular  surface  forming  the  outer  pole  of  the  magnet  should 
be  less  than  1  in.,  it  might  be  necessary,  for  mechanical  reasons, 
to  reduce  the  diameter  in  order  to  obtain  a  practical  design. 
The  total  pull  required  is  4,000  lb.,  or  2,000  Ib.  per  pole.  The 
pull  per  square  inch  of  polar  surface  is,  by  formula  (16)  page  29. 

B2 
Pounds  per  square  inch  =  Y~73QQQ 

whence  the  area  of  each  pole  face  is 


2,000  X  1,730,000 
B2 


If  B  =  6,000  gausses,  A  ==  96  sq.  in.;  and  if  B  =  8,000  gausses, 
A  =  54  sq.  in.     Either  of  these  flux  densities  would  probably 


Magnetising  Coil 


Retaining  Plate 
I  of  (Non-magnetic) 
\  Manganese  Steel, 

with  Stiffening 

Ribs 


FIG.  22. — Circular  lifting  magnet. 

be  suitable  for  a  magnet  of  the  type  considered.  Assuming  a 
minimum  width  of  1  in.  for  the  pole  face  on  the  outer  shell,  we 
have, 

Area  of  outer  ring  =  1  X  TT  X  21 
=  66  sq.  in. 

which  would  provide  the  required  pull  if  B  =  7,240  gausses. 

5 


66  PRINCIPLES  OF  ELECTRICAL  DESIGN 

Let  us  therefore  decide  upon  this  dimension.  The  diameter 
of  the  inner  core  is  obtained  from  the  equation 

j  D2  =  66 

whence  D  —  9.16  in.  It  will  be  better  to.  provide  a  2-in.  hole 
through  the  center  of  the  magnet,  and  have  a  conical  face  to  the 
core,  as  shown  in  sketch.  The  diameter  of  the  central  pole  core 
may  be  10  in.,  and  the  edges  can  be  slightly  bevelled  off  so  that 
the  polar  surface  shall  not  exceed  66  sq.  in. 

In  order  to  introduce  a  factor  of  safety,  and  permit  of  the  iron 
ball  being  lifted  even  when  the  contact  between  magnet  and 
armature  is  imperfect,  the  specification  would  probably  call  for 
a  magnet  powerful  enough  to  attract  the  ball  through  a  distance 
of,  say,  J4  m-  Let  us  further  assume  that,  the  action  being 
intermittent,  the  current  will  flow  through  the  exciting  coil  dur- 
ing only  half  the  time  that  the  magnet  is  in  action.  This  will 
probably  permit  the  use  of  a  current  density  of  1,000  amp.  per 
square  inch  of  copper  section.  Thus,  if  the  winding  space  factor 
may  be  taken  as  0.5,  it  will  be  necessary  to  provide  2  sq.  in.  of 
cross-section  of  coil  for  every  1,000  ampere-turns  of  excitation 
required. 

The  ampere-turns  necessary  to  overcome  the  reluctance  of  the 
double  air  gap  are 

l(SI)a  =  2.025  X  l"g 

=  2.02  X  7,240  X  }4 
=  (say)  8,000,  which  includes  a  small 

allowance  for  the  reluctance  of  the  iron  in  the  circuit.  The 
required  section  of  coil  is  therefore  about  16  sq.  in.  One  of  the 
dimensions  should,  if  possible,  be  kept  within  the  limit  of  3  in. 
in  order  to  avoid  excessive  internal  temperatures.  A  cross- 
section  of  5  in.  by  3  in.  =  15  sq.  in.  will  probably  be  large  enough 
to  accomodate  the  winding. 

The  average  length  per  turn  of  wire  is  7r(10  +  5)  =  47.2  in., 
and  (by  formula  26,  Art.  10,  page  42)  the  cross-section  of 
the  wire,  in  circular  mils,  will  be 

47.2  X  8,000 
(m)  =  -     -ir 

where  E  is  the  voltage  across  the  terminals  of  the  magnet. 
Assuming  this  to  be  120  volts,  the  value  of  (m)  will  be  3,140. 

1  Art.  4,  formula  (5). 


THE  DESIGN  OF  ELECTROMAGNETS  67 

Referring  to  the  wire  table  on  page  34,  the  calculated  size  is  seen 
to  be  only  slightly  less  than  the  cross-section  of  No.  15,  B.  &  S. 
gage.  Using  this  wire,  and  allowing  }-£  in.  for  insulation  be- 
tween the  iron  and  the  coil,  there  will  be  about  68  layers  of  41 
turns,  making  a  total  of,  say,  2,800  turns  in  the  coil.  The  length 
of  wire  will  therefore  be  2,800  X  47.2  -v-  12  =  11,000  ft.,  and 
the  resistance  hot,  i.e.,  at  a  temperature  of  60°C.,  will  be  3.702 
X  11  =  40.6  ohms.  The  current  =  120/40.6  =  2.95  amp.  and 
the  actual  ampere-turns  =  2.95  X  2,800  =  8,260. 

Rise  of  Temperature.  —  The  watts  lost  in  the  field  when  the 
current  is  flowing  are  El  =  120  X  2.95  =  354;  but  since  the 
current  is  supposed  to  be  passing  through  the  windings  during 
only  one-half  the  time  that  the  magnet  is  in  operation,  we  can 
apply  the  rule  referred  to  in  Art.  12,  and  assume  that  the  power  to 
be  dissipated  amounts  to  only  354/2  =  177  watts.  The  total 
surface  of  the  coil  is  47.2(10  +  6)  =  755  sq.  in.;  and  if  we  use  the 
average  value  of  180  for  the  heating  coefficient  fc,  as  suggested 
in  Art.  11  page  46,  the  temperature  rise  will  be 


180  X          =  42.2°C. 


above  the  temperature  of  the  air.  This  figure  is  a  safe  one,  and, 
since  the  iron  shell  offers  a  large  cooling  surface  in  contact  with 
the  air,  it  is  probable  that  the  value  of  the  coefficient  k  in  this 
particular  design  might  be  about  250.  The  temperature  of 
the  windings  will  therefore  not  be  excessive,  and  the  amount  of 
copper  might  even  be  slightly  reduced  if  the  greatest  economy  in 
manufacturing  cost  is  to  be  attained.  Exact  data  for  the  cal- 
culation of  temperatures  in  coils  entirely  surounded  by  iron  are 
not  available,  because  the  thickness  and  radiating  surface  of 
the  external  shell  are  factors  which  will  have  an  appreciable 
influence  on  the  value  of  the  heating  coefficient. 

Calculation  of  Leakage  Flux.  —  In  order  to  provide  sufficient 
cross-section  in  the  magnet,  and  ensure  that  the  flux  density  in 
the  iron  shall  not  be  carried  too  near  the  saturation  limit,  it  is 
necessary  to  estimate  the  amount  of  the  leakage  flux. 

The  permeance  of  the  leakage  paths  may  be  calculated  by 
considering  two  separate  components  of  the  leakage  flux:  (1) 
the  flux  which  passes  between  the  core  and  the  cylindrical  shell 
through  the  space  occupied  by  the  windings,  and  (2)  the  flux 
which  passes  between  the  uncovered  portions  of  the  central  core 


68  PRINCIPLES  OF  ELECTRICAL  DESIGN 

and  the  outer  shell.  Referring  to  Fig.  22,  upon  which  the 
approximate  dimensions  of  the  leakage  paths  have  been  marked, 
the  numerical,  values  of  the  two  permeances  are  seen  to  be, 
approximately, 

3X7r(10  +  5)  X6.45 
for  the  path  (1),  Pi  =  -          5  v  9  54         "  = 

2  X  ir(10  +  5)  X  6.45 
for  path  (2),  P2  =  5.5  x  2.54  •  43.5 

The  flux  through  path  (1)  is 

_  m.m.f. 

^1    —  o  XX    J-    1 


=  374,000 
and  the  flux  through  path  (2)  is, 

$2  =  m.m.f.  X  P2 

=  0.4  TT  X  8,260  X  43.5 
=  450,000 

The  total  leakage  flux  is  $1  +  3>2  =  824,000  maxwells.  The 
useful  flux  is  66  X  6.45  X  7,240  =  3,080,000  maxwells,  and  the 
leakage  factor  is 

3,080,000  X  824,000 
3,080,000 

The  flux  density  in  the  cylindrical  outer  shell,  near  the  yoke, 
will  be  7,240  X  1.27  =  9,200  gausses,  and  this  will  also  be  the 
density  in  the  central  pole  if  the  cross-section  is  the  same,  but  a 
higher  density  would  be  permissible.  The  section  at  every  part 
of  the  magnetic  circuit  can  be  calculated  on  the  basis  of  an 
assumed  density.  As  an  instance,  if  it  is  desired  to  have  a  density 
of  11,000  gausses  in  the  yoke  at  the  section  AB,  the  thickness  of 
the  casting  at  this  point,  as  indicated  by  the  length  of  the  line 
AB,  would  be  obtained  from  the  equation 

3,080,000  X  1.27 
"  A  B  X  TT  X  10  X  6.45 

whence  AB  =  \Y±  in. 

In  this  manner  the  magnetic  circuit  may  be  proportioned. 
The  path  of  the  magnetic  flux  is  indicated  by  the  dotted  lines  in 


THE  DESIGN  OF  ELECTROMAGNETS  69 

Fig.  22,  and  since  the  cross-section  and  length  of  each  part  of  the 
magnetic  circuit  are  now  known,  the  component  of  the  total 
ampere-turns  necessary  to  overcome  the  reluctance  of  the  iron 
or  steel  casting  can  be  calculated  in  the  usual  way.  The  re- 
luctance of  the  steel  sphere  which  constitutes  the  armature 
would  be  considered  negligible  in  these  calculations. 

In  a  well-designed  magnet  of  this  type,  the  reluctance  of  the 
iron  portions  of  the  circuit  is  but  a  small  percentage  of  the  air- 
gap  reluctance,  unless  the  specified  air  gap  is  very  small.  When 
the  armature  is  in  contact  with  the  pole  faces,  the  total  flux  will 
be  greater  than  the  amount  necessary  to  produce  the  required 
initial  pull.  It  is  interesting  and  instructive  to  calculate  the 
pull  between  magnet  and  armature  when  the  air  gap  is  practically 
negligible.  The  limit  is  reached  when  all  the  exciting  ampere- 
turns  are  required  to  overcome  the  reluctance  of  the  iron,  and 
the  calculation  has  to  be  made  by  assuming  probable  values  of  the 
flux  density,  and  then  calculating  the  loss  of  magnetic  potential 
across  each  portion  of  the  circuit. 

With  reference  to  the  important  matter  of  cost;  air-gap  den- 
sities other  than  the  assumed  density  of  7,240  gausses  may  be 
tried  with  a  view  to  obtaining  the  design  of  lowest  first  cost.  A 
saving  in  copper  may  be  effected  by  allowing  the  temperature 
rise  to  approach  as  nearly  as  possible  the  specified  limit;  but  in 
this  as  in  all  economical  designs  of  apparatus  in  which  a  saving 
in  first  cost  is  accompanied  by  a  loss  of  efficiency  in  working,  the 
interests  of  the  user  demand  that  proper  attention  be  paid  to  the 
cost  of  operation  (in  this  case  of  the  I2R  losses)  when  considering 
the  expediency  of  lowering  the  manufacturing  cost  by  econo- 
mizing in  materials. 


CHAPTER  IV 

DYNAMO  DESIGN—  FUNDAMENTAL  CONSIDERATIONS. 
BRIEF  OUTLINE  OF  PROBLEM 

18.  Generation  of  E.m.f.  —  It  has  been  shown  in  previous 
chapters  how  the  strength  and  amount  of  the  magnetic  field 
produced  by  an  electric  current  may  be  calculated,  and  the  next 
step  in  the  development  of  the  dynamo  is  to  consider  how  the 
desired  terminal  voltage  may  be  obtained  by  causing  the  armature 
conductors  to  cut  the  magnetic  flux  which  crosses  the  air  gap  from 
pole  to  armature  core. 

The  D.C.  motor  is  merely  a  dynamo  of  which  the  action  has 
been  reversed;  that  is  to  say,  instead  of  providing  mechanical 
energy  to  drive  the  armature  conductors  through  the  magnetic 
field,  an  electric  current  from  an  outside  source  is  sent  through 
the  armature  winding  which,  by  revolving  in  the  magnetic  field, 
converts  electrical  energy  into  mechanical  energy.  In  the  design 
of  a  D.C.  motor,  the  procedure  is  exctly  the  same  as  for  a  D.C. 
generator,  and  in  the  following  pages  the  dynamo  will  be  thought 
of  mainly  as  a  generator. 

Consider  a  flat  coil  of  insulated  wire  of  resistance  R  ohms, 
consisting  of  S  turns  enclosing  an  area  of  A  square  centimeters. 
Let  this  coil  be  thrust  into,  or  withdrawn  from,  a  magnetic 
field  of  density  B  gausses,  the  direction  of  which  is  normal  to  the 
plane  of  the  coil.  The  quantity  of  electricity  which  will  be  set 
in  motion  is  expressed  by  the  formula 


a  relation  that  can  be  proved  experimentally. 
But  Q=/mXi  =  §X* 

where  t  =  the  time  required  to  enclose  or  withdraw  the  flux 

(*  =  BA), 
Im  =  the  average  value  of  the  current  in  the  coil  during 

this  period,  and 

Em  —  the  average  value  of  the  e.m.f.  causing  the  flow  of 
electricity. 

70 


DYNAMO  DESIGN  71 

Hence  F         (BA)  X  S 

Em  =        — 

where  all  quantities  are  expressed  in  absolute  C.G.S.  units.  If 
we  put  <£  for  the  flux  (BA)  in  maxwells,  and  express  the  e.m.f. 
in  the  practical  system  of  units,  we  have 

Em  =  f£~  volts  (37) 

For  the  condition  S  =  unity,  this  formula  is  clearly  seen  to 
express  the  well-known  relation  between  rate  of  change  of  flux 
and  resulting  e.m.f.,  namely  that  one  hundred  million  maxwells 
cut  per  second  generate  one  volt.  This  is  the  fundamental  law  upon 
which  all  quantitative  work  in  dynamo  design  is  based.  The 
procedure  for  obtaining  a  given  amount  of  flux  was  explained 
in  previous  chapters,  and  we  now  see  that  the  voltage  of  any 
dynamo-electric  generator  may  be  calculated  by  applying  formula 
(37).  For  the  rest,  the  electrical  part  of  the  designer's  work 
consists  in  providing  a  sufficient  cross-section  of  copper  to  carry 
the  required  current,  and  a  sufficient  cross-section  of  iron  to  carry 
the  required  flux,  in  order  that  the  machine  shall  not  heat  ab- 
normally under  working  conditions.  There  are  other  matters  of 
importance  such  as  regulation,  efficiency,  economy  of  material, 
and — in  D.C.  machines — commutation,  which  require  careful 
study;  but  it  is  hardly  an  exaggeration  to  say  that — apart  from 
mechanical  considerations,  which  are  not  dealt  with  in  this  book 
—the  work  of  the  designer  of  electrical  machinery  is  based  on  two 
fundamental  laws:  (1)  the  law  of  the  magnetic  circuit,  namely, 
that  the  flux  is  equal  to  the  ratio  of  magnetomotive  force  to  reluc- 
tance, and  (2)  the  law  of  the  generation  of  an  e.m.f.,  namely,  that 
one  hundred  million  lines  cut  per  second  generate  one  volt. 

At  any  particular  moment  it  is  the  rate  of  change  of  the  flux  in 
the  circuit  that  determines  the  instantaneous  value  of  the 
voltage,  or,  in  symbols, 

d& 
instantaneous  volts  in  circuit  of  one  turn  =  —  -^  X  10~8 

where  the  negative  sign  is  introduced  because  the  developed 
e.m.f.  always  tends  to  set  up  a  current  the  magnetizing  effect  of 
which  opposes  the  change  of  flux. 

Consider 'a  dynamo  with  any  number  of  poles  p.  Fig.  23 
shows  a  four-pole  machine  with  one  face-conductor  driven 


72  PRINCIPLES  OF  ELECTRICAL  DESIGN 

mechanically  at  a  speed  of  N  revolutions  per  minute  through  the 
flux  produced  by  the  field  poles.  If  $  stands  for  the  amount  of 
flux  entering  or  leaving  the  armature  surface  per  pole,  we  may 
write, 

volts  generated  per  conductor  =  1<1 


1<18  V-  an 

.LU     /\  OU 

It  is  not  at  present  necessary  to  discuss  the  different  methods  of 
winding  armatures,  but  let  there  be  a  total  of  Z  conductors 
counted  on  the  face  of  the  armature.  Then,  if  the  connections 
of  the  individual  coils  are  so  made  that  there  are  p\  electrical 
circuits  in  parallel  in  the  armature,  the  generated  volts  will  be 

QpNZ 
=  60  X  Pl  X  108 

This  is  the  fundamental  voltage  equation  for  the  dynamo;  it 
gives  the  average  value  of  the  e.m.f.  developed  in  the  armature 

conductors,  and  since  the  virtual  and 
average  values  are  the  same  in  the 
case  of  continuous  currents,  the 
formula  gives  the  actual  potential 
difference  as  measured  by  a  volt- 
meter across  the  terminals  when  no 
current  is  taken  out  of  the  armature. 
Under  loaded  conditions,  the  e.m.f. 
as  calculated  by  formula  (38)  is  the 
terminal  voltage  plus  the  internal  IR 
FlG  23  pressure  drop. 

The  expression  "  face  conductors  " 

may  be  used  to  define  the  conductors  the  number  of  which  is 
represented  by  Z  in  the  voltage  formula.  It  is  evident  that  this 
number  includes  not  only  the  top  conductors,  but  also  those  that 
may  be  buried  in  the  armature  slots.  The  word  "inductor"  is 
sometimes  used  in  the  place  of  "face  conductor,"  and  where 
either  word  is  used  in  the  following  pages  it  must  be  under- 
stood to  refer  to  the  so-called  "active"  conductor  lying  parallel 
to  the  axis  of  rotation  whether  on  a  smooth  core  or  slotted 
armature. 

19.  The  Output  Formula.  —  The  part  of  the  dynamo  to  be 
designed  first  is  the  armature.  After  the  preliminary  dimensions 
of  the  armature  have  been  determined,  it  is  a  comparatively 


DYNAMO  DESIGN  73 

simple  matter  to  design  a  field  system  to  furnish  the  necessary 
magnetic  flux.  The  designer  is  usually  given  the  following  data: 

Kw.  output, 

Terminal  voltage, 

Speed — revolutions  per  minute.    . 

Sometimes  the  proper  speed  has  to  be  determined  by  the  de- 
signer, as  in  getting  out  a  line  of  stock  sizes  of  some  particular 
type  of  machine;  in  that  case  he  will  be  guided  by  the  practice  of 
manufacturers  and  the  safe  limits  of  peripheral  speed.  Other 
conditions  such  as  temperature  rise,  pressure  compounding, 
sparkless  commutation  of  current,  may  be  imposed  by  specifica- 
tions, but  if  the  designer  can  evolve  a  formula  which  will  give  him 
an  approximate  idea  of  the  weight  or  volume  of  the  armature, 
this  will  be  of  great  assistance  to  him  in  determining  the  leading 
dimensions  for  a  preliminary  design.  Modifications  or  correc- 
tions can  easily  be  made  later,  after  all  the  influencing  factors 
have  been  studied  in  detail.  Many  forms  of  the  output  formula 
are  used  by  designers.  The  formula  is  based  on  certain  broad 
assumptions,  and  is  used  for  obtaining  approximate  dimen- 
sions only.  Attempts  to  develop  exact  output  formulas  of 
universal  application  should  not  be  encouraged  because  it  is  not 
possible  to  include  all  the  influencing  factors.  The  art  of 
designing  will  always  demand  individual  skill  and  judgment, 
which  cannot  be  embodied  in  mathematical  formulas. 

In  developing  an  output  formula  it  is  not  necessary  to  enter  into 
details  of  the  armature  winding,  provided  the  total  number  of 
conductors,  together  with  the  current  and  e.m.f.  in  each,  are 
known. 

Let    3>  =  maxwells  per  pole. 
p  =  number  of  poles. 
N  —  re  volutions 'per  minute. 
Z  =  total  number  of  armature  inductors. 
EC  =  volts  per  conductor. 
Ic  =  amperes  per  conductor. 

The  output  of  the  armature,  expressed  in  watts,  will  be 

W  =  ZEJc  (39) 

where  Ic  should  include  the  exciting  current  in  the  shunt  coils 
of  the  field  winding;  a  refinement  which  need  not,  however, 
enter  in  the  preliminary  work. 


74  PRINCIPLES  OF  ELECTRICAL  DESIGN 

The  voltage  per  conductor  is 

*-e^i  •       (40) 

where  the  unknown  quantity  $  may  be  expressed  in  terms  of  flux 
density  and  armature  dimensions.  Thus 

$p  =  QA5BglawDr  (41) 

where  Bg  =  average  flux  density  in  the  air  gap  under  the  pole 

face.     (Gausses.) 

la  =  gross  length  of  armature  core,  in  inches. 
D  =  diameter  of  armature  core,  in  inches. 

pole  arc 

r  =  the  ratio  —  ^  -  r—  r- 
pole  pitch 

It  will  be  seen  that  the  quantity  la  X  irDr  is  the  area  in  square 
inches  of  the  armature  surface  covered  by  the  pole  shoes;  while 
6.45#0.is  the  flux  in  the  air  gap  per  square  inch  of  polar  surface. 

The  pole  pitch  is  usually  thought  of  as  the  distance  from  center 
to  center  of  pole  measured  on  the  armature  surface;  and  the 
ratio  r  is  therefore  a  factor  by  which  the  total  cylindrical  surface 
of  the  armature  must  be  multiplied  to  obtain  the  area  covered 
by  the  pole  shoes  —  the  effect  of  "fringing"  at  the  pole  tips  being 
neglected. 

Substituting  for  $p  in  equation  (40)  its  value  as  given  by  equa- 
tion (41),  and  putting  this  value  of  Ec  in  equation  (39),  we 
have 


_  gac 

60  X  108 

from  which  it  is  necessary  to  eliminate  Z  and  Ic  if  the  formula 
is  to  have  any  practical  value. 

A  quantity  which  does  not  vary  very  much,  whatever  the 
number  of  poles  or  diameter  of  armature,  is  the  specific  loading, 
which  is  defined  as  the  ampere-conductors  per  inch  of  armature 
periphery.  It  will  be  represented  by  the  symbol  q.  Thus 

ZIC 

q  =  *D 
whence 

ZIC  =  qwD 

Substituting  in  equation  (42),  we  have 
W  = 


DYNAMO  DESIGN 


75 


This  is  not  an  empirical  formula  since  it  is  based  on  fundamental 
scientific  principles,  and  it  is  capable  of  giving  valuable  infor- 
mation regarding  the  size  of  the  armature  core,  provided  the 
quantities  Bgj  q,  and  r,  can  be  correctly  determined. 

The  quantity  Bg  will  depend  somewhat  upon  whether  the 
pole  shoe  is  of  cast  iron  or  steel,  also  upon  the  flux  density  in  the 
armature  teeth,  which,  in  turn,  depends  upon  the  proportions 
of  the  teeth  and  slots.  If  the  flux  density  in  the  teeth  is  very 
high,  this  may  lead  to  (1)  an  excessive  number  of  ampere-turns 
on  the  field  poles  to  overcome  tooth  reluctance,  and  (2)  excessive 
power  loss  in  the  teeth  through  hysteresis  and  eddy  currents. 

As  a  guide  in  selecting  a  suitable  gap  density  for  the  preliminary 
calculations,  the  accompanying  table  may  be  used.  The 
column  headed  Bg  is  the  apparent  air-gap  density  in  gausses, 
while  E" g  is  the  same  quantity  expressed  approximately  in  lines 
per  square  inch. 

APPROXIMATE  VALUES  OF  APPARENT  AIR-GAP  DENSITY 


Output,  kw. 

B,  (gausses) 

B'g  (lines  per  sq.  in.) 

10 

6,300 

41,000 

20 

7,000 

45,000 

30 

7,300 

47,000 

40 

7,600 

49,000 

50 

7,800 

50,000 

100 

8,100 

52,000 

200 

8,500 

55,000 

500  and  larger 

9,000 

58,000 

The  expression  "  apparent  gap  density"  means  that  the  flux  is 
supposed  to  be  distributed  uniformly  over  the  face  of  the  pole 
and  the  effect  of  "fringing"  is  neglected.  Thus 

total  flux  per  pole 
area  of  pole  face 

It  is  customary  to  think  of  this  as  the  average  density  over  the 
armature  surface  covered  by  the  pole  face,  in  which  case 


Ba  = 


°    "   la  XT  X  r 

where  3>,  la)  and  r,  have  the  same  meaning  as  in  formula  (41), 
and  T  is  the  pole  pitch  or  length  of  arc  from  center  to  center  of 
pole  measured  on  the  armature  periphery.  The  lower  values  of 


76 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


Bg  corresponding  to  the  smaller  outputs  are  required  because 
the  increased  taper  of  the  teeth  with  the  smaller  armature 
diameters  would  lead  to  abnormally  high  densities  at  the  root 
of  the  teeth  if  the  air-gap  density  were  not  reduced.  The  figures 
given  in  the  table  are  applicable  to  machines  with  pole  shoes  of 
steel  or  wrought  iron.  If  the  pole  shoes  are  of  cast  iron,  these 
values  should  be  reduced  about  20  per  cent.  Cast-iron  pole 
shoes  are  rarely  used  except  in  very  small  machines. 

The  quantity  q  in  formula  (43)  is  determined  in  the  first  place 
by  the  heating  limits;  but  armature  reaction  and  sparkless 
commutation  have  some  bearing  upon  its  value.  Suitable 
values  of  specific  loading  for  use  in  formula  (43)  may  be  taken 
from  the  accompanying  table. 

APPROXIMATE  VALUES  OF  q 
(Ampere  Conductors  per  Inch  of  Armature  Periphery) 


Kw.  output 

5 

10 

320 

20 

370 

30 

400 

40 

430 

50 

450 

100 

500 

200 

550 

400 

630 

600 

700 

800  and  upward 

760  to  850 

The  quantity  r  in  formula  (43)  usually  has  a  value  between  0.60 
and  0.80,  a  common  value  being  0.70.  When  the  machine  is 

provided  with  commutating  interpoles  the  ratio  — j r— r  must 

be  small  in  order  to  make  room  for  the  interpole.  In  this 
case  the  lower  figure  of  0.60  would  probably  be  selected  as  a 
suitable  value  for  r. 

Approximate  Constants  for  Use  in  Output  Formula. — For  a 
first  approximation,  the  output  formula  (43)  may  be  simplified 
by  substituting  average  values  for  the  quantities  Bg,  q,  and  r. 
Thus,  if  Bg  =  7,500;  q  =  500;  and  r  =  0.7,  the  output  formula 
becomes 

2 
kw.  output  = 


DYNAMO  DESIGN  77 

If  the  speed  of  rotation  (N)  is  not  specified,  it  is  necessary 
to  make  some  assumptions  regarding  the  peripheral  velocity  of 
the  armature.  This  velocity  lies  between  1,200  and  6,000 
ft.  per  minute;  the  lower  values  corresponding  to  machines 
of  which  the  speed  of  rotation  is  low,  while  the  higher  values 
would  be  applicable  to  belt-driven  dynamos,  or  to  direct-coupled 
sets  of  which  the  prime  mover  is  a  high-speed  engine  or  high-head 
waterwheel.  When  the  generator  is  coupled  to  a  steam  turbine, 
the  speed  is  always  exceptionally  high,  and  the  surface  velocity  of 
the  armature  may  then  attain  2  or  3  miles  per  minute.  The 
discussion  of  steam-turbine-driven  generators,  in  so  far  as  the 
electrical  problems  differ  from  those  of  the  lower-speed  machines, 
will  be  taken  up  in  connection  with  alternator  design. 

The  peripheral  velocity  in  feet  per  minute  is, 


TsT 

whence  Ar       ^v 

"s> 

Inserting  this  value  of  N  in  formula  (44)  we  get 

kw.  output  =  (45) 


Relation  of  la  to  D. — The  output  equation  (43)  shows  that  there 
is  a  definite  relation  between  the  volume  of  the  armature  and  the 
output,  provided  the  quantities  represented  by  the  symbols 
Bg,  q,  and  r,  can  be  estimated.  In  order  to  determine  the  rela- 
tion between  the  length  la  and  the  diameter  D,  certain  further 
assumptions  must  be  made.  Thus, 

irDr 
I.  =  -pT  ^         (46) 

where  p  =  the  number  of  poles,  and  k  is  the  ratio  —  -TT-. 

armature  length 

It  is  desirable  to  have  the  pole  face  as  nearly  square  as  possible 
because  this  will  lead  to  the  most  efficient  field  winding.  If  the 
section  of  the  pole  core  departs  considerably  from  the  circular  or 
square  section,  the  length  per  turn  of  field  winding  increases 
without  a  proportionate  increase  of  the  flux  carried  by  the  pole. 
For  a  square  pole  face,  k  =  1  and 

D        p 

—  =  -*-  =  0.45p  (approximately) 

la        Tit 

The  ratio  D/la  usually  lies  between  the  limits  of  0.35p  and  0.65p. 


78  PRINCIPLES  OF  ELECTRICAL  DESIGN 

It  is  not  always  possible  or  desirable  to  provide  a  square  pole 
face,  and  indeed  it  is  necessary  to  check  the  dimensions  of  the 
armature  core  by  calculating  the  peripheral  velocity.  If  a 
suitable  value  for  the  peripheral  velocity  can  be  assumed,  the 
diameter  is  readily  calculated  because 


- 

XX          -  T»T- 

irN 

20.  Number  of  Poles  —  Pole  Pitch  —  Frequency.  —  For  calcula- 
ting the  relation  between  the  length  and  diameter  of  armature  core 
by  formula  (46),  the  number  of  poles  p  must  be  known.  The 
selection  of  a  suitable  number  of  poles  will  be  influenced  by 
considerations  of  frequency  and  pole  pitch. 

Frequency  of  D.C.  Machines.  —  The  frequency  of  currents  in  the 
armature  conductors  and  of  flux  reversals  in  the  armature  core 
generally  lies  between  10  and  40  cycles  per  second  in  continuous- 
current  generators.  Higher  frequencies  are  allowable,  but  should 
be  avoided,  if  possible,  because  —  on  account  of  increased  losses 
in  the  iron,  or  increased  weight  to  limit  these  losses  —  the  use  of 

p       N 
high  frequencies  is  uneconomical.     The  frequency  is  /  =  ^  X  ™ 

whence 


P  -  .  (47) 

This  relation  is  useful  for  determining  the  probable  number  of 
poles  when  the  diameter,  and  therefore  the  peripheral  velocity, 
are  not  known. 

Pole  Pitch.  —  The  width  of  the  pole  pitch  is  limited  by  armature 
reaction.  It  will  readily  be  understood  that  the  armature 
ampere-turns  per  pole  will  be  proportional  to  the  pole  pitch, 
except  for  variations  in  the  specific  loading  (q)  .  With  a  large 
number  of  ampere-turns  per  pole  on  the  armature,  it  is  necessary 
to  provide  a  correspondingly  strong  exciting  field  in  order  that  the 
armature  shall  not  overpower  the  field  and  produce  excessive 
distortion  of  the  air-gap  flux,  resulting  in  poor  regulation  and 
sparking  at  the  brushes  with  changes  of  load.  A  good  practical 
rule  is  that  the  ampere-conductors  on  the  armature  shall  not 
exceed  15,000  per  pole;  i.e.}  in  the  space  of  one  pole  pitch. 

Ampere-turns  on  Armature.  —  Exactly  what  is  meant  by  the 
expression  "ampere-turns  per  pole"  when  applied  to  the  arma- 
ture winding  should  be  clearly  understood.  In  a  two-pole 


DYNAMO  DESIGN 


79 


machine,  the  total  number  of  ampere-turns  on  the  armature  is 

Z  Z 

SI  —  2  X  Ic  because  the  current  Ic  in  -~  conductors   on   one- 
half  of  the  armature  surface  is  balanced  by  an  equal  but  opposite 


FIG.  24. — Current  distribution — bi-polar  armature. 


current  in  -~  conductors   on   the   other   half   of   the   armature 

surface,  as  indicated  in  Fig.  24.     Now,  since  there  are  two  poles, 
we  may  say  the  ampere-turns  per  pole  are 

1*7  T 
£J  1  c 

2  ~2~ 


OS/), 


or  just  half  the  number  of  ampere-conductors  in  a  pole  pitch. 
This  rule  applies  also  to  the  multipolar  machines.     Thus,  in  Fig. 


Direction  of  Motion  of  Armature 
Surface 

FIG.  25. — Armature  m.m.f. — multipolar  dynamo. 

25,  the  horizontal  datum  line  may  be  thought  of  as  the  developed 
surface  of  a  four-pole  dynamo  armature.  The  brushes  are  so 
placed  on  the  commutator  that  they  short-circuit  the  coils  when 
these  are  approximately  halfway  between  the  pole  tips.  It  is, 
therefore,  permissible  to  show  the  brushes  in  this  diagram  as  if 


80  PRINCIPLES  OF  ELECTRICAL  DESIGN 

they  were  actually  in  contact  with  the  conductors  on  the  "  geo- 
metric neutral"  line.  The  armature  m.m.f.  will  always  be  a 
maximum  at  the  point  where  the  brushes  are  placed,  because  the 
direction  of  the  current  in  the  conductors  changes  at  this  point, 
producing  between  the  brushes  belts  of  ampere-conductors  of 
opposite  magnetizing  effect,  as  indicated  in  Fig.  25.  The  broken 
straight  line  indicates  the  distribution  of  the  armature  m.m.f. 
over  the  surface.  Its  maximum  positive  value  occurs  at  A 
and  its  maximum  negative  value  at  B.  These  maximum  ordi- 
nates  are  of  the  same  height,  and  equal  to  one-half  the  ampere- 
conductors per  pole  pitch  as  will  be  readily  understood  by  inspect- 
ing the  diagram.  Thus,  whether  the  machine  is  bipolar  or 
multipolar,  the  armature  ampere-turns  per  pole  are 

(SI)  a  =  ~  ampere-conductors  per  pole  pitch 

- 1  <« 

In   practice,   a  safe  limit  for  the  pole  pitch  is  r  =  - 

If  q  =  750,  the  maximum  allowable  pole  pitch  is  r  =  20  in., 
which  dimension  is  rarely  exceeded  in  ordinary  types  of  dynamos. 
Number  of  Poles. — The  formula  (45)  gives  the  output  in  terms 
of  the  peripheral  velocity.  In  its  complete  form  it  would  be 
written 

kw.  output  =  laDv  X  Bgqr  X  4  X  1Q-11  (49) 

By  eliminating  la  and  D  from  the  equation,  it  is  possible  to 
arrive  at  an  expression  for  the  number  of  poles  in  terms  of  the 
peripheral  velocity  and  other  quantities  for  which  values  can 
be  assumed.  Thus,  by  (46) 

_  TirD 
'  ~pk~ 

In  order  to  eliminate  D,  let  (SI)a  be  the  armature  ampere- 
turns  per  pole.  Then 

=  *D  =  2(SI)a 

p  q 

whence  ^  _  2p(SI)a 

irq 

By  substituting  these  values  for  la  and  D  in  equation  (49)  we 
get 

.        vy     irkq  X  1011  ,_n, 

P  =  kw*  X  «  (50) 


DYNAMO  DESIGN 


81 


By  assuming  values  for  the  quantities  in  the  right-hand  side 
of  the  equation,  a  reasonable  figure  for  the  number  of  poles  can 
be  obtained.  As  an  example,  let  the  assumed  values  be  as 
follows, 

Jb-1 

q  =  650 
r  =  0.7 
v  ==  4,500 
Ba  =  7,500 
(SI)a  =  7,500 

The  number  of  poles  will  then  be 

p  =  0.0138  kw. 

If  the  machine  is  to  have  an  output  of  500  kw.,  the  estimated 
number  of  poles  is 

p  =  7 

and  since  an  even  number  of  poles  is  necessary,  the  required 
figure  is  6  or  8.  This,  of  course,  would  mean  a  change  in  one 
or  more  of  the  assumed  quantities. 

The  proper  number  of  poles  is  determined  partly  by  the  amount 
of  the  current  to  be  collected  from  each  brush  set.  This  will 
influence  the  selection  of  suitable  values  for  k  and  (SI)a  in 
formula  (50).  Values  as  high* as  1,000  amp.  per  brush  arm  are 
used  in  connection  with  low-voltage  machines;  but,  on  machines 
wound  for  250  to  500  volts,  the  current  collected  per  brush  arm 
usually  lies  between  the  limits  of  700  and  300  amp. 

As  a  guide  in  selecting  a  suitable  number  of  poles  for  a  pre- 
liminary design,  the  accompanying  table  may  be  of  use.  It  is 
based  on  the  usual  practice  of  manufacturers. 

NUMBER  OF  POLES  AND  USUAL  SPEED  LIMITS  OF  DYNAMOS 


Output,  kw. 

No.  of  poles 

Speed,   rev.   per   min. 

Oto       10 

2 

2,400  to  600 

10  to       50 

4 

1,300  to  350 

50  to     100 

4  or    6 

1,100  to  230 

100  to     300 

6  or    8 

700  to  160 

300  to     600 

6  to  10 

500  to  120 

600  to  1,000 

8  to  12 

400  to  100 

1,000  to  3,000 

10  to  20 

200  to    70 

82  PRINCIPLES  OF  ELECTRICAL  DESIGN 

When  using  this  or  any  other  table  or  data  intended  to  assist 
the  designer  with  approximate  values,  it  is  necessary  to  exercise 
judgment,  or  at  least  be  guided  by  common  sense.  For  instance 
it  may  be  necessary  to  depart  from  the  values  given  in  the  table 
in  the  case  of  machines  direct-coupled  to  slow-running  engines. 
This  is  especially  worth  noting  in  the  case  of  the  smaller 
machines,  which  may  require  more  than  four  poles  in  order  to 
give  the  best  results  on  very  low  speeds. 


CHAPTER  V 
ARMATURE  WINDINGS  AND  SLOT  INSULATION 

21.  Introductory. — The  object  of  this  chapter  is  to  explain 
the  essential  points  which  the  designer  must  keep  in  mind  when 
determining  the  number  of  slots,  the  space  taken  up  by  insula- 
tion, the  cross-section  of  the  copper  windings,  and  the  method 
of  connecting  the  individual  conductors  so  as  to  produce  a  finished 
armature  suitable  for  the  duty  it  has  to  perform.  It  is  assumed 
that  the  reader  is  familiar  with  the  appearance  of  a  D.C.  machine 
and  understands  generally  the  function  of  the  commutator. 
For  this  reason  it  is  proposed  to  omit  such  elementary  descrip- 
tive matter  as  may  be  found  in  every  textbook  treating  of  elec- 
trical machinery.  On  the  other  hand,  the  practical  details  of 
manufacture  and  much  of  the  nomenclature  used  in  the  design 
room  and  shops  of  .manufacturers  will  also  be  omitted,  because 
space  does  not  permit  of  the  subject  being  treated  exhaustively; 
but  if  the  reader  will  exercise  his  judgment  and  rely  upon  his 
common  sense,  he  will  be  able  to  design  a  practical  armature 
winding  to  fulfil  any  specified  conditions. 

The  direction  of  the  generated  e.m.f.  will  depend  upon  the 
direction  of  the  flux  through  which  the  individual  conductor  is 
moving,  and  it  is  therefore  a  simple  matter  so  to  connect  the 
armature  coils  that  the  e.m.fs.  shall  be  additive.  It  does  not 
matter  whether  the  machine  is  bipolar  or  multipolar,  ring-  or 
drum-wound,  it  is  always  possible  to  count  the  number  of  con- 
ductors in  series  between  any  pair  of  brushes  and  thus  make 
sure  that  the  desired  voltage  will  be  obtained. 

Closed-coil  windings  will  alone  be  considered,  because  the 
open-circuit  windings — as  used  in  the  early  THOMSON-HOUSTON 
machines  and  other  generators  for  series  arc  lighting  systems — 
are  now  practically  obsolete.  Another  type  of  machine,  known 
as  the  homopolar  or  acyclic  D.C.  generator,  although  actually 
used  and  built  at  the  present  time,  has  a  limited  application 
and  will  not  be  considered  here.  The  absence  of  the  com- 
mutator is  the  feature  which  distinguishes  this  machine  from  the 

83 


84  PRINCIPLES  OF  ELECTRICAL  DESIGN 

more  common  types;  but  it  is  not  suitable  for  high  voltages, 
and  the  friction  and  PR  losses  in  the  brushes  are  very  large.1 

22.  Ring-  and  Drum-wound  Armatures. — The  GRAMME  ring 
winding  is  now  practically  obsolete.     In  this  type  of  machine 
the  coils  form  a  continuous  winding  around  the  armature  core 
which  is  in  the  form  of  a  ring  with  a  sufficient  opening  to  allow 
of  the  wires  passing  down  the  inside  of  the  core  parallel  to  the 
axis  of  rotation.     The  objections  to  this  winding  are  the  high 
resistance  and  reactance  of  the  armature  coils  due  to  the  large 
proportion  of  the  " inactive"  material  per  turn.     This  has  a 
bearing  not  only  on  the  cost  and  efficiency  of  the  machine  but 
also  on  commutation,  because  the  high  inductance  of  the  wind- 
ings is  liable  to  cause  sparking  at  the  brushes. 

In  the  drum  winding  all  the  conductors  are  on  the  outside 
surface  of  the  armature,  and  although  space  is  usually  provided, 
even  in  small  machines,  between  the  inside  of  the  core  and  the 
shaft,  this  space  is  used  for  ventilating  purposes  only  and  is  not 
occupied  by  the  armature  coils.  The  drum  winding  is  simply 
the  result  of  so  arranging  the  end  connections  that  the  e.m.fs. 
generated  in  the  various  face  conductors  shall  assist  each  other 
in  providing  the  total  e.m.f.  between  brushes.  Both  windings, 
whether  of  the  ring  or  drum  type,  are  continuous  and  closed 
upon  themselves,  and  if  the  brushes  are  lifted  off  the  commutator 
there  will  be  no  circulating  currents  because  the  system  of 
conductors  as  a  whole  is  cutting  exactly  the  same  amount  of 
positive  as  of  negative  flux. 

23.  Multiple    and    Series    Windings. — Nearly    all    modern 
continuous-current  generators  are  provided  with  former-wound 
coils.     These  coils  are  made  up  of  the  required  number  of  turns, 
and  pressed  into  the  proper  shape  before  being  assembled  in 
the  slots  of  the  armature  core.     Smooth-core  armatures  are  very 
rarely  used,  and  the  slotted  armature  alone  will  be  considered. 
In  all  but  two-pole  machines  (which  are  rarely  made  except  for 
small  outputs  or  exceptionally  high  speeds)  there  is  practically 
only  one  type  of  coil  in  general  use.     This  is  generally  of  the 

1  For  information  on  the  homopolar  type  of  machine  see  "  Acyclic 
(Homopolar)  Dynamos,"  by  J.  E.  NOEGGERATH,  Trans.  A.  I.  E.  E.,  vol. 
XXIV  (1905),  pp.  1  to  18,  and  the  article  by  the  same  writer,  "Acyclic 
Generators,"  in  the  Electrical  World,  Sept.  12,  1908,  p.  575.  Also  "Homo- 
polar  Generators, "  by  E.  W.  Moss  and  J.  MOULD,  Journal  Inst.  E.  E.,  vol. 
49  (1912),  pp.  804  to  816. 


ARMATURE  WINDINGS  AND  SLOT  INSULA TION    85 


shape  shown  in  Fig.  26.  The  finished  coil  may  consist  of  any 
number  of  turns,  but  as  there  are  two  coil-sides  in  each  slot, 
there  will  be  an  even  number  of  inductors  per  slot.  The  portion 
of  the  armature  periphery  spanned  by  each  coil  is  approximately 
equal  to  the  pole  pitch  T,  because  it  is  necessary  that  one  coil 
side  shall  be  cutting  positive  flux  while  the  other  coil  side  is 
cutting  negative  flux.  If  the  two  coil  sides  lie  in  slots  exactly 


\ 


FIG.  26. — Form-wound  armature  coil. 

one  pole  pitch  apart,  the  winding  is  said  to  be  full  pitch.  If 
the  width  of  coil  is  less  than  r,  the  winding  is  said  to  be  short 
pitch  or  chorded.  The  shortening  of  the  pitch  slightly  reduces 
the  length  of  inactive  copper  in  the  end  connections.  Some 
designers  claim  that  it  gives  better  commutation;  but,  on  the 
other  hand,  it  reduces  the  effective  width  of  the  zone  of  commu- 
tation, and  it  is  doubtful  if  either  winding  has  a  distinct 
advantage  over  the  other. 

There  are  two  entirely  different  methods  of  joining  together 


86 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


the  individual  coils.  Thus,  if  the  two  ends  of  the  coil  shown  in 
Fig.  26  are  connected  to  neighboring  commutator  bars,  as  in 
Fig.  27a,  we  obtain  a  multiple  or  lap  winding,  while,  if  the  ends 
are  taken  to  commutator  bars  approximately  a  pole  pitch  apart, 
(Fig.  276),  we  obtain  the  series  or  wave  winding.  The  important 
distinction  between  these  two  styles  of  winding  is  the  fact  that, 
with  the  multiple  winding,  there  are  as  many  sets  of  brushes  and 


(a) 


(b) 


[< r  (  Approximately  ) —    — >-J 

FIG.  27. — Multiple  and  series  armature  coil  connections. 

as  many  parallel  paths  through  the  armature  as  there  are  poles 
while,  with  the  series  winding,  there  are  only  two  electrical  paths 
in  parallel  through  the  armature,  and  only  two  sets  of  brushes  are 
necessary,  although  a  greater  number  of  brush  sets  may  be  used.1 

1  What  are  known  as  simplex  windings  are  here  referred  to.  Multiplex 
windings  may  be  used  on  lap-wound  machines  when  the  current  is  large  and 
it  is  desired  to  have  two  or  more  separate  circuits  connected  in  parallel  by 
sufficiently  wide  brushes.  This  tends  to  improve  commutation.  In  series- 
wound  machines  the  multiple  winding  has  the  advantage  that  it  enables 
the  designer  to  obtain  more  than  two  circuits  in  parallel,  but  not  as  many 
as  there  are  poles.  In  this  manner  it  is  possible  to  provide  for  a  number  of 
parallel  paths  somewhere  between  the  limits  set  by  the  simplex  wave  and 
lap  windings  respectively.  These  windings  are,  however,  rarely  met  with. 
Again,  a  duplex  winding  may  be  singly  re-entrant  or  doubly  re-entrant. 
In  the  former  case,  the  winding  would  close  on  itself  only  after  passing  twice 
around  the  armature,  while,  in  the  latter  case,  there  would  be  two  inde- 
pendent windings.  It  is  suggested  that  the  reader  need  not  concern 
himself  with  these  distinctions,  which  have  no  bearing  on  the  principles  of 
armature  design.  More  complete  information  can  be  found  in  many  text- 
books and  in  the  handbooks  for  electrical  engineers. 


ARM  A  TURE  WINDINGS  AND  SLOT  INSULA  TION    87 

The  two  kinds  of  winding  are  shown  diagrammatically  in 
Figs.  28  and  29.  The  former  shows  a  simplex  lap-wound 
multipolar  drum  armature,  while  the  latter  represents  in  a 
similar  diagrammatic  manner  a  simplex  wave-wound  multipolar 
drum  armature.  In  practice  there  would  ordinarily  be  a  greater 
number  of  coils  and  commutator  bars,  and  the  conductors  would 
be  in  slots.  This  is  indicated  by  the  grouping  of  the  conductors 


FIG.  28. — Diagram  of  simplex  multiple  winding. 

in  pairs,  the  even-numbered  inductors  being  (say)  in  the  bottom 
of  the  slot,  with  the  odd-numbered  inductors  immediately 
above  them  in  the  top  of  the  slot. 

It  should  particularly  be  noted  that  the  lap  or  multiple  wind- 
ing provides  as  many  electrical  circuits  in  parallel  as  there  are 
poles,  and  the  number  of  brush  sets  required  is  the  same  as  the 
number  of  poles.  Thus  if  7  is  the  current  in  the  external  circuit, 
plus  the  small  component  required  for  the  shunt  field  excitation, 
the  current  in  the  armature  windings  is  I/p  and  the  current 

21 

collected  by  each  set  of  brushes  is  — 

P 


88 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


In  the  case  of  the  wave,  or  series,  winding  there  will  be  two 
paths  in  parallel  through  ^he  armature,  whatever  may  be  the 
number  of  poles.  Two  sets  of  brushes  will,  therefore,  suffice 
to  collect  the  current;  but  more  sets  may  be  used  if  desired,  in 
order  to  reduce  the  necessary  length  of  the  commutator.  In 
this  case  the  brushes  would  be  placed  one  pole  pitch  apart 
around  the  commutator,  and  all  brush  sets  of  the  same  polarity 
would  be  joined  in  parallel.  This  does  not,  however,  increase 


FIG.  29. — Diagram  of  simplex  series  winding. 


the  number  of  electrical  paths  in  parallel  in  the  armature,  but 
merely  facilitates  the  collection  of  the  total  current,  as  will  be 
understood  by  carefully  studying  the  winding  diagrams.  The 
series  winding  is  usually  adopted  when  the  voltage  is  high  and 
the  current  correspondingly  reduced;  it  is,  therefore,  rarely  neces- 
sary to  provide  more  than  two  sets  of  brushes.  When  a  greater 
number  of  brush  sets  is  provided  on  a  series-wound  machine, 
it  is  not  easy  to  ensure  that  the  current  will  be  equally  divided 
between  the  various  sets  of  brushes.  What  is  known  as  selective 


ARMATURE  WINDINGS  AND  SLOT  INSULATION    89 


commutation  then  occurs,  each  brush  set  collecting  current  in 
proportion  to  the  conductance  of  the  brush  contact.  This  leads 
to  sparking  troubles  unless  ample  brush  surface  is  provided. 

The  fact  that  there  may  be  only  two  sets  of  brushes  on  a  multi- 
polar  dynamo  does  not  necessarily  indicate  a  wave-wound 
armature.  The  commutators  of  lap-wound  machines  are  some- 
times provided  with  an  internal  system  of  cross-connections 
whereby  all  commutator  bars  of  the  same  potential  are  joined 


s 


(a)  (6) 

FIG.  30. — Appearance  of  lap  and  wave  windings. 

together.  This  allows  of  only  two  sets  of  brushes  being  used; 
but  the  length  of  the  commutator  must,  of  course,  be  increased 
to  provide  the  brush-contact  surface  necessary  for  the  proper 
collection  of  the  current.  The  external  appearance  of  the 
parallel  and  series  windings  respectively  is  indicated  by  sketches 
(a)  and  (6)  of  Fig.  30.  The  observer  is  supposed  to  be  looking 
down  on  the  cylindrical  surface  of  the  finished  armature. 

If  there  are  two  coil  sides  in  each  slot,  the  number  of  commu- 
tator bars  will  be  the  same  as  the  number  of  armature  slots, 
whether  the  coils  are  connected  to  form  a  multiple  or  a  series 
winding.  It  is,  however,  by  no  means  necessary  to  limit  the 


90  PRINCIPLES  OF  ELECTRICAL  DESIGN 

number  of  commutator  bars  to  the  number  of  slots,  as  the  total 
number  of  inductors  in  each  slot  may  be  subdivided,  and  a 
correspondingly  greater  number  of  commutator  bars  can  be 
used.  This  point  will  be  again  referred  to  when  treating  of  the 
slot  insulation. 

With  a  series-wound  armature,  the  number  of  commutator 
bars  cannot  be  a  multiple  of  the  number  of  poles,  because  this 
would  lead  to  a  closed  winding  after  stepping  once  around  the 
armature  periphery.  The  winding  must  advance  or  retrogress 
by  one  commutator  bar  when  it  has  been  once  around  the  arma- 
ture, and  this  leads  to  the  rule  that  a  series-wound  machine  must 
have  a  number  of  commutator  segments  such  as  to  fulfil  the 
condition : 

Number  of  commutator  segments  1         ,p 

in  wave-wound  machine  j          2  " 

where  k  is  any  whole  number.1 

24%  Equalizing  Connections  for  Multiple -wound  Armatures. — 
If  the  magnetic  circuits  of  the  various  parallel  paths  in  the  lap- 
wound  dynamo  are  not  of  equal  reluctance,  there  will  be  an  un- 
balancing of  the  generated  e.m.fs.  producing  circulating  currents 
through  the  brushes.  The  inequality  of  the  magnetic  perme- 
ances is  usually  due  to  excentricity  of  the  armature  relatively 
to  the  bore  of  the  poles,  and  even  when  the  unbalancing  effect  is 
small  in  a  new  machine,  it  is  liable  to  increase  owing  to  wear  of 
the  bearings. 

Fig.  31  is  a  developed  view  of  a  four-pole  winding.  Sup- 
pose that  the  section  A  of  the  armature  winding  is  nearer  to  the 
pole  face  than  the  section  C.  The  voltage  generated  in  the  con- 
ductors occupying  the  latter  position  will  be  less  than  in  the 

1  Although  an  armature  may  be  provided  with  an  odd  number  of  slots,  it 
does  not  follow  that  it  will  accommodate  a  wave  winding  suitable  for  all 
voltages,  without  modification.  Thus,  if  a  six-pole  machine  has  73  armature 
slots,  it  may  be  necessary  to  have  two  coils  (or  four  coil-sides)  per  slot  in 
order  to  obtain  the  necessary  voltage  and  avoid  too  great  a  difference  of 
potential  between  adjacent  commutator  bars.  This  means  that  the  number 
of  coils  and  of  commutator  segments  would  be  146,  which  would  ^not  be 
suitable  for  a  wave  winding.  By  having  one  dummy  or  "dead"  coil,  the 
total  number  of  coils  (and  commutator  segments)  will  be  145,  which  being 
equal  to  (48  X  %)  +1  will  give  a  wave  winding.  The  "dead"  coil  is  put 
in  for  appearance  and  to  balance  the  armature,  but  it  is  not  connected  up. 
The  use  of  dead  coils  should  be  avoided  as  it  adds  to  the  difficulties  of 
commutation. 


ARMATURE  WINDINGS  AND  SLOT  INSULATION    91 

conductors  moving  through  the  region  A.  The  result  will  be 
a  tendency  for  a  current  to  circulate  in  the  path  AECF  as 
indicated  by  the  dotted  arrows.  The  net  result  will  be  a  strength- 
ening of  the  current  leaving  the  machine  at  one  set  of  brushes 
and  a  corresponding  weakening  of  the  current  at  the  other  set 
of  brushes.  This  unbalancing  of  the  current  may  lead  to  serious 
sparking  troubles.  To  prevent  the  inequality  of  voltage  in  the 
different  sections  of  the  windings  it  is  necessary  to  go  to  the  root 
of  the  trouble  and  correct  the  differences  in  the  reluctance  of  the 
various  magnetic  paths.  This  cannot,  however,  always  be  ac- 
complished perfectly  or  in  a  lasting  manner;  but,  by  providing 


"~ 



h  —  -r 

—  -i 

— 

1 

---rl^-^-J 

A 

1 

B 

C 

D 

FIG.  31. — Equalizer  connections. 

easy  paths  for  the  out-of-balance  current  components,  it  is 
possible  to  equalize  the  differences  of  pressure  before  the  current 
reaches  the  brushes.  This  is  done  by  connecting  together  points 
on  the  armature  winding  which  should  be  at  the  same  potential. 
In  practice  a  number  of  insulated  copper  rings  are  provided  and 
connected  to  equipotential  points  on  the  commutator.  The 
dotted  lines  in  Fig.  31  show  six  equalizing  rings.  Actually, 
from  six  to  eight  points  per  pair  of  poles  would  probably  be  cross- 
connected. 

It  should  be  clearly  understood  that  the  equalizing  connections 
of  lap-wound  armatures  do  not  prevent  the  unbalancing  of 
currents;  but,  by  providing  a  short-circuit  to  the  paths  through 
brushes  and  connecting  leads  of  the  same  sign,  they  tend  to 
maintain  the  equality  of  currents  through  the  various  brush 
sets.  In  the  simplex  wave  winding,  with  only  two  armature 
paths  in  parallel,  equalizing  connections  are  not  necessary. 

26.  Insulation  of  Armature  Windings. — No  great  amount  of 
insulation  is  necessary  on  each  wire  or  conductor  of  an  armature 


92  PRINCIPLES  OF  ELECTRICAL  DESIGN 

winding  because,  even  in  machines  of  large  output,  the  voltage 
generated  per  turn  of  wire  is  comparatively  small.  The  diff- 
erence of  potential  between  the  winding  as  a  whole  and  the 
armature  core  may,  however,  be  very  great  on  high-voltage 
machines,  and  the  slot  lining  must  be  designed  to  withstand 
this  pressure  with  a  reasonable  factor  of  safety.  About  the 
same  amount  of  insulation  as  will  be  necessary  for  the  slot 
linings  will  also  have  to  be  provided  between  the  upper  and 
lower  coil-sides  in  each  slot,  because  the  potential  difference 
between  the  two  sets  of  conductors  in  the  slot  is  the  same  as  that 
between  the  terminals  of  the  machine.  The  space  occupied  by 
insulation  relatively  to  the  total  space  available  for  the  winding 
will  depend  not  only  upon  the  voltage  of  the  dynamo,  but  also 
upon  such  factors  as  the  number  of  slots  and  their  cross-section 
and  proportions.  Even  if  the  total  slot  area  remains  constant, 
the  larger  number  of  slots  will  naturally  require  the  greater 
amount  of  insulation,  and  thus  reduce  the  space  available  for 
copper.  Again,  a  wide  slot,  by  reducing  the  tooth  width,  may 
be  the  cause  of  unduly  high  densities  in  the  teeth,  while  a  deep 
slot  is  undesirable  on  account  of  increased  inductance  of  the 
windings,  and  because  it  may  lead  to  an  appreciably  reduced 
iron  section  at  the  root  of  the  tooth  in  armatures  of  small 
diameter. 

With  the  ordinary  double-layer  winding,  the  square  coil 
section  would  give  the  best  winding  space  factor.  Thus  if  each 
of  the  two  coil-sides  were  made  of  square  cross-section,  the  total 
depth  of  slot  including  space  for  binding  wires  or  wedge  would 
be  from  two  and  one-fourth  times  to  two  and  one-half  times  the 
width.  In  practice  the  slot  depth  is  frequently  three  times  the 
width,  but  this  ratio  should  not  exceed  3J^  because  the  design 
would  be  uneconomical,  and  the  high  inductance  of  the  winding 
might  lead  to  commutation  difficulties. 

Although  the  calculation  of  flux  densities  in  the  teeth  will  be 
dealt  with  later,  it  may  be  stated  that  it  is  usual  to  design  the 
slot  with  parallel  sides  and  make  the  slot  width  from  0.4  to  0.6 
times  the  slot  pitch.  It  is  very  common  to  make  slot  and  tooth 
width  the  same  (i.e.,  one-half  the  pitch)  on  the  armature  surface, 
especially  in  small  machines.  In  large  machines  the  ratio 
tooth  width  . 

•asTridth- 1S 

What  has  been  referred  to  as  the  slot  pitch  may  be  denned  as 


ARMATURE  WINDINGS  AND  SLOT  INSULA TION    93 

the  ratio,  armature  surface  periphery  divided  by  the  total  number 
of  slots. 

26.  Number  of  Teeth  on  Armature. — It  is  obvious  that  a  small 
number  of  teeth  would  lead  to  a  reduction  of  space  taken  up  by 
insulation  and,  generally  speaking,  would  lead  also  to  a  saving 
in   the   cost   of   manufacture.     Other   considerations,   however, 
show  that  there  are  many  points  in  favor  of  a  large  number  of 
teeth.     Unless  the  air  gap  is  large  relatively  to  the  slot  pitch, 
there  will  be  appreciable  eddy-current  loss  in  the  pole  pieces  on 
account  of  the  tufting  of  the  flux  lines  at  the  tooth  top.     Again, 
pulsations  of  flux  in  the  magnetic  circuit  are  more  liable  to  be  of 
appreciable  magnitude  with  few  than  with  many  teeth,  and  when 
the  tooth  pitch  is  wide  in  relation  to  the  space  between  pole  tips, 
commutation  becomes  difficult  because  of  the  variation  of  air-gap 
reluctance  in  the  zone  of  the  commutating  field.     A  good  practi- 
cal rule  is  that  the  number  of  slots  per  pole  shall  not  be  less  than 
10,  and  that  there  shall  be  at  least  three  and  one-half  slots  in  the 
space  between  pole  tips.     In  high-speed  machines  with  large  pole 
pitch,  from  14  to  18  slots  per  pole  would  usually  be  provided. 
With  the  exception  of  small  generators  (machines  with  armatures 
of  small  diameter),  the  cross-section  of  the  slot  is  about  constant, 
and  approximately  equal  to  1  sq.  in.     This  corresponds  to  about 
1,000  amp.  conductors  per  slot  for  machines  up  to  600  volts, 
on  the  basis  of  the  current  densities  to  be  discussed  later. 

27.  Number  of  Commutator  Segments — Potential  Difference 
between  Segments. — Machines  may  be  built  with  a  number  of 
commutator  bars  equal  to  the  number  of  slots  in  the  armature 
core.     In  this  case  there  will  be  one  coil  per  slot,  i.e.,  two  coil- 
sides  in  each  slot.     There  is,  however,  no  reason  why  the  number 
of  coils1  should  not  be  greater  than  the  number  of  slots.     The 
usual  number  of  commutator  segments  per  slot  is  two  or  three 
in  low-voltage  machines,  with  a  maximum  of  four  or  five  in  low- 
speed  dynamos  for  high  voltages.     The  number  of  commutator 
bars  may  therefore  be  a  multiple  of  the  number  of  slots.     A 

1  The  word  coil  as  here  used  denotes  the  number  of  turns  included  between 
the  tappings  taken  to  commutator  bars.  In  practice  one-half  the  number  of 
conductors  in  a  slot  might  be  taped  up  together  and  handled  as  a  single  coil, 
but  if  tappings  are  taken  from  the  ends  so  as  to  divide  the  complete  coil  in 
two  or  more  sections  electrically,  we  may  speak  of  four,  six,  or  more  coil- 
sides  in  a  single  slot,  notwithstanding  the  fact  that  these  "may  be  bunched 
together  and  treated  as  a  unit  when  placing  the  finished  coils  in  position. 


94  PRINCIPLES  OF  ELECTRICAL  DESIGN 

large  number  of  bars  improves  commutation,  but  increases  the 
cost  of  the  machine;  a  large  diameter  of  commutator  is  necessary 
in  order  that  the  individual  sector  shall  not  be  too  thin.  The 
copper  bars  are  insulated  from  each  other  by  mica,  usually 
about  ^2  m-  thick,  increasing  to  J^o  m-  f°r  machines  of  1,000 
volts  and  upward.  It  follows  that  a  commutator  with  a  very 
large  number  of  segments  is  less  easily  assembled  and  less  satis- 
factory from  the  mechanical  standpoint  than  one  with  fewer 
segments. 

The  best  way  to  determine  the  proper  number  of  commutator 
bars  for  a  particular  design  of  dynamo  is  to  consider  the  voltage 
between  neighboring  bars.  This  voltage  is  variable,  and  depends 
upon  the  distribution  of  the  magnetic  flux  over  the  armature 
surface,  and  upon  the  position  of  the  armature  coil  under  con- 
sideration. The  maximum  potential  difference  between  adjacent 
commutator  bars  rarely  exceeds  40  volts,  and  the  average  voltage 
should  be  considerably  lower  than  this.  The  average  voltage 
between  bars  may  be  defined  as  the  potential  difference  between 
+  and  -  -  brush  sets  divided  by  the  number  of  commutator 
segments  counted  between  the  brushes  of  opposite  sign.  As  a 
rough  guide,  it  may  be  stated  that  the  value  of  15  volts  (average) 
between  segments  should  not  be  exceeded  in  machines  without 
interpoles.  About* double  this  value  is  permissible  as  an  upper 
limit  on  machines  with  commutating  interpoles,  especially  if 
they  are  provided  with  compensating  pole-face  windings  which 
prevent  the  distortion  of  flux  distribution  under  load.  In 
practice  the  allowable  average  voltage  between  commutator 
bars  is  based  upon  the  machine  voltage  and,  to  some  extent, 
upon  the  kilowatt  output,  although  few  designers  appear  to 
pay  much  attention  to  the  influence  of  the  current  in  determining 
the  number  of  commutator  segments.  As  an  aid  to  design,  the 
following  values  may  be  used  for  the  purpose  of  deciding  upon 
a  suitable  number  of  coils  and  commutator  bars. 

Machine  voltage  Volts  between  commutator  segments 

110  1      to    6 

220  2.5  to  10 

600  5      to  18 

1,200  9      to  25 

28.  Nature  and  Thickness  of  Slot  Insulation. — Since  the 
average  voltage  between  the  terminals  of  an  armature  coil  does 
not  exceed  25  volts,  it  follows  that  the  potential  difference  be- 


ARMATURE  WINDINGS  AND  SLOT  INSULATION       95 

tween  the  conductors  in  one  coil  cannot  be  very  high.  The 
copper  conductors  are  usually  insulated  with  cotton  spun  upon 
the  wire  in  two  layers.  Cotton  braiding  is  sometimes  used  on 
large  conductors  of  rectangular  section;  and  a  silk  covering  is 
used  on  very  small  wires  where  a  saving  of  space  may  be  effected 
and  an  economical  design  obtained  notwithstanding  the  high 
price  of  the  silk  covering.  A  triple  cotton  covering  is  occasionally 
used  when  the  potential  difference  between  turns  exceeds  20 
volts.  Conductors  of  large  cross-section  may  be  insulated  by  a 
covering  of  cotton  tape  put  on  when  the  coil  is  being  wound. 

In  addition  to  the  comparatively  small  amount  of  insulation 
on  the  wires,  a  substantial  thickness  of  insulation  must  be  pro- 
vided between  the  armature  core  and  the  winding  as  a  whole. 
The  materials  used  for  slot  lining  are: 

1.  Vulcanized  Fiber;  Leatheroid  or  Fish  Paper;  Manilla  Paper; 
Pressboard;  Presspahn;  Horn  Fiber;  etc.;  all  of  which,  being 
tough  and  strong,  are  used  mainly  as  a  mechanical  protection 
because  they  are  more  or  less  hygroscopic  and  cannot  be  relied 
upon  as   high-pressure   insulators,  especially  when  moisture  is 
present. 

2.  Mica;  Micanite;  Mica  Paper  or  Cloth.     These  materials 
are  good  insulators,  and  the  pure  mica  or  the  micanite  sheet  will 
withstand  high  temperatures.     Sheet  micanite  is  built  up  of 
small  pieces  of  mica  split  thin  and  cemented  together  by  varnish. 
The  finished  sheet  is  subjected  to  great  pressure  at  high  tem- 
peratures in  order  to  expel  the  superfluous  varnish.     Mica  is 
hard  and  affords  good  protection  against  mechanical  injury; 
but  it  is  not  suitable  for  insulating  corners  or  surfaces  of  irregular 
shape. 

3.  Treated    Fabrics,    such    as    Varnished    Cambric;    Empire 
Cloth,  etc.     These  provide  a  means  of  applying  a  good  insulat- 
ing protection  to  coils  of  irregular  shape.     Linseed  oil  is  very 
commonly  used  in  the  preparation  of  these  insulating  cloths 
and  tapes  because  it  has  good  insulating  properties  and  remains 
flexible  for  a  very  long  time. 

It  is  common  practice  to  impregnate  the  finished  armature 
coil  with  an  insulating  compound,  and  press  it  into  shape  at  a 
fairly  high  temperature.  When  this  is  done,  ordinary  untreated 
cotton  tape  is  used  in  place  of  the  varnished  insulation. 

Fig.  32  shows  a  typical  slot  lining  for  a  500-volt  winding. 
This  can,  however,  be  modified  in  many  respects,  and^the  reader 


96 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


Hard  Wood  Wedge 


(Slot  Lining  of  Press- 
N  pahn  about  0.010  " 
<  Thick 

v  Cotton  Tape;  Empire 
Xiloth  or  Micanite  to 
Total  Radial  Thick- 
ness about  0.030" 
Cotton  Covering; 
Braiding  or  Tape  on 
Conductors,  of  0.014" 
to  0.025"Single 
Thickness 

\Pres8pnhn  or  Fish 
Paper  Dividing  Strip, 
of  about  O.Olo'Tbick 


h  Paper  at  Bottom 
of  Slot  about  0.020 >J 
Thick 


need  not  at  present  concern  himself  with  practical  details  of 
manufacture.  It  is  obvious  that  the  insulation  should  be  so 
arranged  as  to  leave  the  greatest  possible  amount  of  space  for 

the  copper.  The  insulation 
may  be  placed  around  the  in- 
dividual coils,  or  in  the  slot 
before  the  coils  are  inserted. 
If  preferred,  part  of  the  insu- 
lation may  be  put  around  the 
coils  and  the  remainder  in 
the  form  of  a  slot  lining.  The 
essential  thing  is  to  have 
sufficient  thickness  of  insula- 
tion between  the  cotton-cov- 
ered wires  and  the  sides  of  the 
slot.  The  following  figures 
FIG.  32. — Insulation  of  conductors  in  may  be  used  in  determining 

the  necessary  thickness  of  slot 

lining.  These  figures  give  the  thickness,  in  inches,  of  one  side 
only,  and  this  is  also  the  thickness  that  should  be  provided  be- 
tween the  upper  and  lower  coil  sides  in  the  slot. 

For  machines  up  to  250  volts 0 . 035  in. 

For  machines  up  to  500  volts 0 . 045  in. 

For  machines  up  to  1,000  volts 0. 06    in. 

For  machines  up  to  1,500  volts 0.075  in. 

In  high-voltage  machines  an  air  space  is  sometimes  allowed 
between  the  end  connections,  i.e.,  the  portions  of  the  coils  not 
included  in  the  slots.  This  air  clearance  would  be  from  %  to 
K  in.  for  a  difference  of  potential  of  1,000  volts,  with  an  addition 
of  y$  in.  for  every  500  volts.  In  regard  to  the  surface  leakage 
where  the  coils  pass  out  from  the  slots;  a  breakdown  of  insulation 
at  this  point  is  usually  guarded  against  by  allowing  the  slot 
lining  to  project  at  least  J-£  in.  beyond  the  end  of  the  slot.  For 
working  pressures  above  500  volts,  add  J^  in.  for  every  additional 
500  volts. 

The  finished  armature  should  withstand  certain  test  pressures 
to  ensure  that  the  insulation  is  adequate.  The  standardization 
rules  of  the  A.I.E.E.  call  for  a  test  pressure  of  twice  the  normal 
voltage  plus  1,000  volts. 

29.  Current  Density  in  Armature  Conductors. — The  per- 
missible current  density  in  the  armature  coils  is  limited  by  tern- 


ARMATURE  WINDINGS  AND  SLOT  INSULATION       97 

perature  rise.  The  hottest  accessible  part  of  the  armature, 
after  a  full-load  run  of  sufficient  duration  to  attain  very  nearly 
the  maximum  temperature,  should  not  be  more  than  40°  or  45°C. 
above  the  room  temperature.  No  definite  rules  can  be  laid 
down  in  the  matter  of  armature  conductor  section  because  the 
ventilation  will  be  better  in  some  designs  than  in  others,  and  a 
large  amount  of  the  heat  to  be  dissipated  from  the  armature  core 
is  caused  by  the  iron  loss  which,  in  turn,  depends  upon  the  flux 
density  in  teeth  and  core. 

The  current  density  in  the  armature  windings  generally  lies 
between  the  limits  of  1,500  and  3,000  amp.  per  square  inch.  If 
the  armature  were  at  rest,  the  permissible  current  density  would 
be  approximately  inversely  proportional  to  the  specific  loading, 
q,  i.e.,  to  the  ampere-conductors  per  inch  of  armature  periphery. 
When  the  armature  is  rotating,  the  additional  cooling  effect  due 
to  the  movement  through  the  air  will  be  some  function  of  the 
peripheral  velocity,  and,  for  speeds  up  to  about  a  mile  a  minute — 
or,  say,  6,000  ft.  per  minute — the  permissible  increase  of  current 
density  will  be  approximately  proportional  to  the  increase  in 
speed.  The  constants  for  use  in  an  empirical  formula  expressing 
these  relations  are  determined  from  tests  on  actual  machines, 
and  the  writer  proposes  the  following  formula  for  use  in  deciding 
upon  a  suitable  current  density  in  the  armature  winding: 


where  A  stands  for  amperes  per  square  inch  of  copper  cross- 
section,  and  v  is  the  peripheral  velocity  in  feet  per  minute. 

30.  Length  and  Resistance  of  Armature  Winding. — Before  the 
resistance  drop  and  the  PR  losses  in  the  armature  can  be  calcu- 
lated, it  is  necessary  to  estimate  the  length  of  wire  in  a  coil. 
This  length  may  be  considered  as  made  up  of  two  parts:  (1)  the 
"active"  part,  being  the  straight  portion  in  the  slots,  and  (2) 
the  end  connections. 

The  appearance  of  the  end  connection  is  generally  as  shown  in 
Fig.  33,  and  since  the  pitch  of  the  coil  is  measured  on  the  cir- 
cumference of  the  armature  core,  the  sketch  actually  represents 
the  coils  laid  out  flat,  before  springing  into  the  slots. 

The  angle  a  which  the  straight  portion  of  the  end  connections 

makes  with  the  edge  of  the  armature  core  is  sin"1  >  where 
X  is  the  slot  pitch;  s,  the  slot  width;  and  5,  any  necessary  clear- 

7 


98 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


ance  between  the  coils.  This  clearance  need  be  provided  only 
in  high-pressure  machines,  or  when  it  is  desired  to  improve 
ventilation.  For  approximate  calculations  on  machines  up  to 
600  volts,  the  angle  a  may  be  calculated  from  the  relation 
1.15s 


sin  a 


In  practice  the  angle  a  usually  lies  between  35  and  40  degrees. 
The  length  of  the  straight  part  AC  (Fig.  33)  is  — —  where  BA 


COS  a 


T    . 


is  half  the  coil  pitch,  or  x  in  the  case  of  a  full-pitch  winding. 


FIG.  33.  —  End  connections  of  armature  coil. 

The  portion  of  the  end  connections  between  the  end  of  the 
slot  and  the  beginning  of  the  straight  portion  AC  is  about  % 
in.  in  low-voltage  machines,  increasing  to  1  in.  in  machines 
for  pressures  between  500  and  1,000  volts.  The  allowance  for 
the  loop  where  the  coil  is  bent  over  to  provide  for  the  lower  half 
of  the  coil  clearing  the  upper  layer  of  conductors  will  depend 
upon  the  depth  of  the  coil-side  and  therefore  upon  the  depth  of 
the  slot.  If  d  is  the  total  slot  depth,  in  inches,  an  allowance  equal 
to  2d  will  be  sufficient  for  this  loop.  The  total  length  of  coil 
outside  the  slots  of  a  low-voltage  armature  will  therefore  be 


where  r  must  be  taken  to  represent  the  coil  pitch  instead  of  the 
pole  pitch  if  the  winding  is  of  the  chorded  or  short-pitch  type. 


ARMATURE  WINDINGS  AND  SLOT  INSULATION       99 

The  average  length  per  turn  of  one  coil  will  be  2la  -f  le  where  la 
is  the  gross  length  of  the  armature  core.  A  small  addition  should 
be  made  for  the  connections  to  the  commutator,  especially  if 
the  coil  has  few  turns.1  The  resistance  of  each  coil,  and  there- 
fore of  each  electrical  path  through  the  armature,  may  now  be 
readily  calculated.  In  arriving  at  the  resistance  of  the  armature 
as  a  whole  it  is  important  to  note  carefully  the  number  of  coils 
in  series  in  each  armature  path,  and  the  number  of  paths  in 
parallel  between  the  terminals  of  the  machine.  As  a  check  on 
the  calculated  figures,  the  IR  drop,  or  the  PR  loss,  in  the  arma- 
tures of  commercial  machines,  expressed  as  a  percentage  of  the 
terminal  voltage  or  of  the  rated  output,  as  the  case  may  be,  is 
usually  as  stated  below: 

In    10-kw.  dynamo 3 . 1  to  3 . 8  per  cent. 

In    30-kw.  dynamo 2.6to3.2  per  cent. 

In    50-kw.  dynamo 2.4to3.0  per  cent. 

In  100-kw.  dynamo 2 . 1  to  2 . 6  per  cent. 

In  200-kw.  dynamo 2.0  to  2.4  per  cent. 

In  500-kw.  dynamo 1 . 9  to  2 . 1  per  cent. 

1  There  is  another  type  of  end  connection,  known  as  the  involute  end 
winding.  It  is  not  much  used ;  but  those  interested  in  the  matter  are  referred 
to  the  first  volume  of  "The  Dynamo  "  by  HAWKINS  and  WALLIS  (  WHITTAKER 
&  Co.),  where  the  manner  of  calculating  the  length  of  these  end  con- 
nections is  explained. 


CHAPTER  VI 

LOSSES  IN  ARMATURES— VENTILATION- 
TEMPERATURE  RISE 

31.  Hysteresis  and  Eddy-current  Losses  in  Armature  Stamp- 
ings.— The  loss  due  to  hysteresis  in  iron  subjected  to  periodical 
reversals  of  flux  may  be  expressed  by  the  formula, 

watts  per  pound  =  KhBl-*f 

where  KH  is  the  hysteresis  constant  which  depends  upon  the 
magnetic  qualities  of  the  iron.  The  symbols  B  and  /  stand  as 
before  for  the  flux  density  and  the  frequency. 

An  approximate  expression  for  the  loss  due  to  eddy  currents 
in  laminated  iron  is, 

watts  per  pound  =  Ke  (Bft)* 

where  t  is  the  thickness  of  the  laminations,  and  Ke  is  a  constant 
which  is  proportional  to  the  electrical  conductivity  of  the  iron. 

With  the  aid  of  such  formulas,  the  hysteresis  and  eddy-current 
losses  can  be  calculated  separately  and  then  added  together  to 
give  the  total  watts  lost  per  pound.  This  method  will  give  good 
results  in  the  case  of  transformers;  but  when  the  reversals  of 
flux  are  due  to  a  rotating  magnetic  field,  as  in  dynamo-electric 
machinery,  the  losses  do  not  follow  the  same  laws  as  when  the 
flux  is  simply  alternating;  and  moreover  there  are  many  causes 
leading  to  losses  in  built-up  armature  cores  which  cannot  easily 
be  calculated.  These  additional  losses  include  eddy  currents  due 
to  burrs  on  the  edges  of  stampings  causing  metallic  contact 
between  adjacent  plates.  There  are  also  eddy  currents  produced 
in  the  armature  stampings  due  to  the  fact  that  the  flux  cannot 
everywhere  be  confined  to  a  direction  parallel  to  the  plane  of 
the  laminations.  Some  flux  enters  the  armature  at  the  two 
ends  and  also  into  the  sides  of  the  teeth  through  the  spaces  pro- 
vided for  ventilation.  Since  this  flux  enters  the  iron  in  a  direction 
normal  to  the  plane  of  the  laminations  it  is  sometimes  account- 
able for  quite  appreciable  losses.  For  these  reasons  calculations 

100 


LOSSES  IN  ARMATURES  101 

of  core  losses  should  be  based  on  the  results  of  tests  conducted 
with  built-up  armatures  rotated  in  fields  of  known  strength. 
Such  tests  are  made  at  different  frequencies,  and  the  results, 
plotted  in  graphical  form,  give  the  total  watts  lost  per  pound  of 
iron  stampings  at  different  flux  densities,  a  separate  curve  being 
drawn  for  each  frequency.  The  reader  is  referred  to  the  hand- 
books of  electrical  engineers  for  useful  data  of  this  sort;  but  for 
approximate  calculations  of  core  losses,  the  total  iron  loss  per 
cycle  may  be  considered  constant  at  all  frequencies.  This 
assumption  allows  of  a  single  curve  being  plotted  to  show  the 
connection  between  watts  lost  per  pound  and  the  product  kilo- 
gausses  X  cycles  per  second.  This  has  been  done  in  Fig.  34 
which  is  based  on  experiments  conducted  by  MESSRS.  PARSHALL 
and  HobART  and  confirmed  lately  by  PROFESSORS  ESTERLINE 
and  MOORE  at  Purdue  University.  The  curve  gives  average 
losses  in  commercial  armature  iron  stampings  0.014  in.  thick. 
Great  improvements  in  the  magnetic  qualities  of  dynamo 
and  transformer  iron  have  been  brought  about  during  the  last 
20  years,  and  the  introduction  of  3  to  4  per  cent,  of  silicon  in 
the  manufacture  of  the  material  known  as  silicon  steel  has 
given  us  a  material  in  which  not  only  the  hysteresis,  but  also 
the  eddy-current  losses,  have  been  very  considerably  lowered. 
There  are  great  variations  of  quality  in  armature  stampings, 
and  values  obtained  from  Fig.  34  would  not  be  sufficiently 
reliable  for  the  use  of  the  commercial  designer  of  any  but  small 
machines.  By  taking  pains  in  assembling  in  stampings  to 
avoid  burrs  and  short-circuits  between  adjoining  plates,  the 
total  iron  loss  may  be  considerably  reduced.  In  large  machines, 
with  a  surface  which  is  small  in  proportion  to  the  volume, 
the  losses  will  usually  be  less  than  would  be  indicated  by  Fig. 
34.  In  the  absence  of  reliable  tests  on  machines  built  with  a 
particular  quality  of  iron  punchings,  it  is  suggested  that  the 
values  obtained  from  Fig.  34  may  be  reduced  as  much  as  50  per 
cent,  in  cases  where  extra  care  and  expense  with  a  view  to  re- 
ducing losses  are  justified;  and  for  silicon  steel  (a  more  costly 
material  than  the  ordinary  iron  plates)  the  reduction  may  be  as 
much  as  70  per  cent. 

When  calculating  the  watts  lost  in  the  armature  core,  it  is 
necessary  to  consider  the  teeth  independently  of  the  section 
below  the  teeth.  This  is  because  the  flux  density  in  the  teeth 
is  not  the  same  as  that  in  the  body  of  the  armature. 


102          PRINCIPLES  OF  ELECTRICAL  DESIGN 

The  calculation  of  the  watts  lost  in  the  core  below  the  teeth 
is  a  simple  matter  provided  the  assumption  can  be  made  that 
the  flux  density  has  the  same  value  at  all  points.  Although  in- 
correct, this  assumption  is  very  commonly  made;  and,  for  the 
purpose  of  estimating  the  rise  in  temperature,  the  flux  density 
may  be  calculated  by  dividing  half  the  total  flux  per  pole  by  the 
net  cross-section  of  the  armature  core  below  the  teeth.  A 
reference  to  Fig.  34  will  give  the  watts  per  pound,  and  this, 


17 

16 
15 
14 
13 
12 
11 

"Sio 

1    9 

Is 

«7 

^    6 
5 
4 
3 
2 
1 
0 

/ 

/ 

S 

/ 

/ 

/ 

/ 

* 

/ 

/ 

/ 

/ 

'  ' 

/ 

/ 

For  Carefully  Assembled 
High  Grade  Iron  Stampings 
Multiply  the  Watts  Obtained 
from  Curve  by  0.5 

For  Silicon  Steel  Stampings 
Multiply  the  Watts  Obtained 
from  Curve  by  0.3 

j 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

^ 

^ 

0          100         200        300        400        500         600        700         800       900       1000      1100 
Kilogausses  x  Frequency,  or    ^- 

FIG.  34. — Losses  in  armature  stampings. 

when  multiplied  by  the  total  weight  of  iron  in  the  core  (excluding 
the  weight  of  the  teeth),  will  be  the  approximate  total  loss  due 
to  hysteresis  and  eddy  currents. 

In  order  to  calculate  the  losses  in  the  armature  teeth,  it  is 
necessary  to  know  exactly  what  is  the  flux  density  at  all  sections 
of  the  tooth.  This  is  not  readily  calculated,  because  some  of  the 
flux  from  the  pole  pieces  enters  the  armature  through  the  sides 
of  the  teeth  and  the  bottom  of  the  slots.  Again,  in  the  case 
of  armatures  of  small  diameter  having  teeth  of  which  the 
taper  may  be  considerable,  the  change  of  section  alters  the  flux 
density  and  the  degree  of  saturation,  so  that  it  is  almost  im- 
possible to  determine  accurately  the  average  value  of  the  tooth 


LOSSES  IN  ARMATURES  103 

density  for  use  in  calculating  the  watts  lost.  The  question  of 
flux  density  in  the  teeth  will  be  again  referred  to  when  discussing 
the  m.m.f.  necessary  to  provide  the  required  flux;  and  formulas 
will  be  developed  for  use  in  calculating  the  actual  flux  density 
in  the  iron  of  the  teeth.  For  the  purpose  of  estimating  the  tem- 
perature rise,  we  shall  assume  that  the  whole  of  the  flux  from 
each  pole  enters  the  armature  through  the  teeth  under  the  pole 
(the  effect  of  fringing  at  pole  tips  being  neglected);  and  if  the 
teeth  are  not  of  uniform  section  throughout  their  length,  the 
average  section  will  be  used  for  calculating  the  flux  density. 
Thus,  let 

3>  =  the  total  flux  per  pole, 
r  ==  pole  pitch, 
X  =  tooth  pitch, 
t  =  width  of  tooth  at  center, 
ln  =  net  length  of  iron  in  armature, 

pole  arc 

r  =  ratio  -^ rr-r- 

pole  pitch 

then  the  number  of  teeth  under  each  pole  is  r  -  and  the  flux  per 

A 

4>X 
tooth  is  —     The  flux  density  in  the  tooth,  on  the  assumptions 

4>X 
previously  made,  would,  therefore,  be     -,-  gausses  if  the  dimen- 

TTlLn 

sions  are  expressed  in  centimeters.  By  referring  to  the  curve 
Fig.  34,  the  watts  lost  in  the  teeth  per  pound  of  iron  can  be  found. 
The  length  ln  is  simply  the  gross  length  of  the  armature  core  less 
the  space  taken  up  by  ventilating  ducts  and  insulation  between 
armature  stampings.  The  question  of  vent  ducts  will  be  taken 
up  immediately;  but,  even  when  a  suitable  allowance  has  been 
made  for  the  spaces  between  the  assembled  sections  of  the 
armature  core,  a  further  correction  must  be  made  to  allow 
for  the  thin  paper  or  other  insulation  between  the  stampings. 
The  space  taken  up  by  this  insulation  will  vary  between  7  and 
10  per  cent,  of  the  total  space.  Thus,  if  la  is  the  gross  length 
of.  the  armature,  and  lv  the  total  width  of  all  vent  ducts,  the 
net  length  of  the  armature  core  would  be. 

ln  =  0.92  (la  -  lv) 

if  the  space  occupied  by  the  insulation  between  laminations  is 
8  per  cent. 


104          PRINCIPLES  OF  ELECTRICAL  DESIGN 

32.  Usual  Densities  and  Losses  in  Armature  Cores. — The 

flux  density  in  the  core  below  the  teeth  will  be  determined  by 
considerations  of  heating  and  efficiency.  The  same  may  be 
said  of  the  tooth  density,  but  in  this  case  the  total  weight  of  iron 
is  relatively  small,  and  higher  densities  are  permissible.  It  is 
desirable  to  have  a  high  flux  density  in  the  teeth  because  this 
leads  to  a  "  stiff er"  field  and  reduces  the  distortion  of  air-gap 
flux  distribution  caused  by  the  armature  current.  Better  pres- 
sure regulation  is  thus  obtained,  and  also  improved  commutation, 
especially  on  machines  without  interpoles  where  the  fringe  of 
flux  from  the  leading  pole  tip  is  used  for  reversing  the  e.m.f. 
in  the  short-circuited  coils  under  the  brush.  If  the  density  in 
the  teeth  is  forced  to  very  high  values,  the  losses  will  be  excessive, 
especially  if  the  frequency  is  also  high;  another  disadvantage 
being  the  large  magnetizing  force  necessary  to  overcome  the 
reluctance  of  the  teeth  and  slots. 

The  accompanying  table  gives  flux  densities  in  teeth  and  core 
that  are  rarely  exceeded  in  ordinary  designs  of  continuous-current 
machines. 

UPPER  LIMITS  OF  FLUX  DENSITY  IN  DYNAMO  ARMATURES  (GAUSSES) 


Frequency,  / 

Density   in   teeth 

Density  in  core 

10 

23,000 

15,000 

20 

22,000 

14,000 

30 

21,000 

13,000 

40 

20,000 

12,000 

As  a  guide  to  the  permissible  losses  in  the  armature  punch- 
ings  of  D.C.  machines,  the  following  figures  will  be  useful. 
They  are  based  on  modern  practice  and  should  not  be  greatly 
exceeded  if  the  efficiency  and  temperature  rise  are  to  be  kept 
within  reasonable  limits. 

Output  of  machine  Core  loss,  expressed  as  percentage 

of  output 

10  kw 2. 8  to  3. 3 

20  kw 2.5  to  3.0 

50  kw 2.0  to  2. 4 

100  kw 1.5  to  1.8 

500  kw 1 . 3  to  1 . 5 

1,000  kw 1 . 2  to  1 . 4 


LOSSES  IN  ARMATURES  105 

33.  Ventilation  of  Armatures. — Recent  improvements  in 
dynamo-electric  machinery  have  been  mainly  along  the  lines 
of  providing  adequate  means  by  which  the  heat  due  to  power 
losses  in  the  machine  can  be  carried  away  at  a  rapid  rate, 
thus  increasing  the  maximum  output  from  a  given  size  of 
frame. 

Still  air  is  a  very  poor  conductor  of  heat;  but  when  a  large 
volume  of  cool  air  is  passed  over  a  heated  surface,  it  will  effectu- 
ally reduce  the  temperature  which  may,  by  this  means,  be  kept 
within  safe  limits. 

The  rotation  of  the  armature  of  an  electric  generator  will  pro- 
duce a  draught  of  air  which  may  be  sufficient  to  carry  away  the 
heat  due  to  PR  and  hysteresis  losses  without  the  aid  of  a  blower 
or  fan.  Self-ventilating  machines  are  less  common  at  the  present 
time  than  they  were  a  few  years  ago;  but,  by  providing  a  suffi- 
cient number  of  suitably  proportioned  air  ducts  in  the  body 
of  the  armature,  machines  of  moderate  size  may  still  be  built 
economically  without  forced  ventilation.  Radial  ducts  are 
provided  by  inserting  special  ventilating  plates  at  intervals 
of  2  to  4  in.,  and  so  dividing  the  armature  core  into  sections 
around  which  the  air  can  circulate.  The  width  of  these  venti- 
lating spaces  (measured  in  a  direction  parallel  to  the  axis  of 
rotation)  is  rarely  less  than  %  in.  or  more  than  %  in.  in  machines 
without  forced  ventilation.  A  narrower  opening  is  liable  to 
become  choked  up  with  dust  or  dirt,  while  the  gain  due  to  a 
wider  opening  is  very  small,  and  does  not  compensate  for  the 
necessary  increase  in  gross  length  of  armature.  The  ventilating 
plates  usually  consist  of  iron  stampings  similar  in  shape  to  the 
armature  stampings,  but  thicker.  Radial  spacers  of  no  great 
width,  but  of  sufficient  strength  to  resist  crushing  or  bending, 
are  riveted  to  the  flat  plates;  they  are  so  spaced  as  to  coincide 
with  the  center  of  each  tooth,  and  allow  the  air  to  pass  outward 
by  providing  a  number  of  small  openings  on  the  cylindrical 
surface  of  the  armature. 

Openings  must  also  be  provided  between  the  shaft  and  the 
inside  bore  of  the  armature  through  which  the  cool  air  may  be 
drawn  to  the  radial  ventilating  ducts.  The  radial  spacers  on 
the  ventilating  plates  assist  the  passage  of  the  air  through  the 
ducts,  their  function  being  similar  to  that  of  the  vanes  in  a 
centrifugal  fan.  Apart  from  the  ventilating  ducts,  the  outer 
cylindrical  surface  of  the  armature  is  effective  in  getting  rid  of  a 


106          PRINCIPLES  OF  ELECTRICAL  DESIGN 

large  amount  of  heat,  and  the  higher  the  peripheral  velocity  of 
the  armature,  the  better  will  be  the  cooling  effect. 

When  forced  ventilation  is  adopted,  a  fan  or  centrifugal  blower 
may  be  provided  at  one  end  of  the  armature.  This  may  as- 
sist the  action  of  radial  ventilating  ducts,  or  it  may  draw  air 
through  axial  ducts.  When  axial  air  ducts  are  provided,  the 
radial  ventilating  spaces  are  omitted,  and  the  gross  length  of  the 
armature  may  therefore  be  reduced.  The  ventilation  is  through 
holes  punched  in  the  armature  plates  which,  when  assembled, 
will  provide  a  number  of  longitudinal  openings  running  parallel 
with  the  armature  conductors.  These  openings  may  be  circular 
in  section  and  should  not  be  less  than  1  in.  in  diameter,  especially 
when  the  axial  length  of  the  armature  is  great,  because  they  will 
otherwise  offer  too  much  resistance  to  the  passage  of  the  air, 
and  will  also  be  liable  to  become  stopped  up  with  dirt. 

One  advantage  of  axial  ducts — which,  however,  can  only  be 
used  with  forced  ventilation — is  that  the  heat  from  the  body 
of  the  armature  can  travel  more  easily  to  the  surface  from  which 
the  heat  is  carried  away  than  in  the  case  of  radial  ducts.  When 
the  cooling  is  by  radial  ducts,  the  heat  due  to  the  hysteresis  and 
eddy-current  losses  in  the  core  must  travel  not  only  through  the 
iron,  which  is  a  good  heat  conductor,  but  also  through  the  paper 
or  other  insulation  between  laminations,  which  is  a  poor  conductor 
of  heat.  As  a  rough  approximation,  it  may  be  said  that  the 
thermal  conductivity  of  the  assembled  armature  stampings  is 
fifty  times  greater  in  the  direction  parallel  to  the  plane  of  the 
laminations  than  in  a  direction  perpendicular  to  this  plane.  For 
this  reason,  radial  vent  ducts,  to  be  effectual,  must  be  provided 
at  frequent  intervals.  The  thickness  of  any  one  block  of  stamp- 
ings between  radial  vent  ducts  rarely  exceeds  3  in. 

The  coefficients  for  use  in  calculating  temperature  rise  are 
based  on  data  obtained  from  actual  machines,  and  owing  to 
variations  in  design  and  proportions  they  are  at  the  best  un- 
reliable. When  forced  ventilation  is  adopted — whether  with 
radial  or  axial  vent  ducts — it  is  possible  to  design  the  fans 
or  blowers  to  pass  a  given  number  of  cubic  feet  of  air  per  second, 
and  the  quantity  can  readily  be  checked  by  tests  on  the  finished 
machine.  The  design  of  such  blowers  does  not  come  within  the 
scope  of  this  book,  neither  is  it  possible  to  discuss  at  length  the 
whole  subject  of  ventilation  and  temperature  rise.  For  a  more 
complete  study  of  this  problem,  the  reader  is  referred  to  other 


LOSSES  IN  ARMATURES  107 

sources  of  information.  A  very  good  treatment  of  the  subject 
will  be  found  in  Chaps.  IX  and  Xof  PROFESSOR  MILES  WALKER'S 
recently  published  book  on  dynamo-electric  machinery.1 

A  good  practical  rule  for  estimating  the  quantity  of  air  nec- 
essary to  carry  away  the  heat  when  forced  ventilation  is  used 
is  based  on  the  fact  that  a  flow  of  1  cu.  ft.  of  air  per  minute  will 
carry  heat  away  at  the  rate  of  0.536T  watts,  where  T  is  the 
number  of  degrees  Centigrade  by  which  the  temperature  of  the 
air  has  been  increased  while  passing  over  the  heated  surfaces. 
Thus,  if  the  difference  in  temperature  between  the  outgoing 
and  incoming  air  is  not  to  exceed  19°C.,  it  will  be  necessary  to 
provide  at  least  100  cu.  ft.  of  air  per  minute  for  each  kilowatt 
lost  in  the  machine. 

The  power  required  to  drive  the  ventilating  fan  is  not  very 
easily  estimated  as  it  depends  upon  the  velocity  of  the  air 
through  the  passages.  The  velocity  of  the  air  through  the  ducts 
and  over  the  cooling  surfaces  is  usually  from  2,000  to  4,000  ft. 
per  minute  and  should  preferably  not  exceed  5,000  ft.  per  minute; 
with  higher  velocities  the  friction  loss  might  be  excessive. 

As  a  very  rough  guide  to  the  power  required  to  drive  the 
ventilating  fans,  the  following  figures  may  be  useful : 

For  50-kw.  dynamo,  150  watts. 
For  200-kw.  dynamo,  500  watts. 
For  1,000-kw.  dynamo,  2,000  watts. 

34.  Cooling  Surfaces  and  Temperature  Rise  of  Armature. — 
Specifications  for  electrical  machinery  usually  state  that  the 
temperature  rise  of  any  accessible  part  shall  not  exceed  a  given 
amount  after  a  full-load  run  of  about  6  hr.  duration.  The 
permissible  rise  of  temperature  over  that  of  the  surrounding  air 
will  depend  upon  the  room  temperature.  It  usually  lies  between 
40°  and  50°C.  The  surface  temperature  is  actually  of  little 
importance  and  is  no  indication  of  the  efficiency  of  a  machine, 
but,  by  keeping  the  surface  temperature  below  a  specified  limit, 
the  internal  temperatures  are  not  likely  to  be  excessive,  and 
the  durability  of  the  insulation — upon  which  the  life  of  the  ma- 
chine is  largely  dependent — will  thereby  be  ensured.  The  de- 
signer must,  however,  see  that  ventilating  ducts  or  surfaces  are 
provided  at  sufficiently  frequent  intervals  to  allow  of  the  heat 

1  MILES  WALKER:  "Specification  and  Design  of  Dynamo-electric  Ma- 
chinery," LONGMANS,  GREEN  &  Co. 


108          PRINCIPLES  OF  ELECTRICAL  DESIGN 

being  carried  away  without  requiring  very  great  differences  of 
temperature  between  the  internal  portions  of  the  material 
where  the  losses  occur  and  the  surfaces  in  contact  with  the  air. 
The  thermal  conductivity  of  all  materials  used  in  construction, 
and  of  the  combinations  of  these  materials,  must  be  known  before 
accurate  calculations  can  be  made  on  the  internal  temperatures; 
but,  as  an  indication  of  how  the  insulation  tends  to  prevent  the 
passage  of  the  heat  to  the  cooling  surfaces,  the  following  figures 
are  of  interest.  The  figures  in  the  column  headed  "Thermal 
conductivity"  express  the  heat  flow  in  watts  per  square  inch  of 
cross-section  for  a  difference  of  1°C.  between  parallel  faces  1 
in.  apart. 

Material  Thermal    conductivity 

Steel  punchings,  along  laminations  1 . 6 
Steel  punchings,  across  laminations 

(8  per  cent,  paper  insulation) 0 . 038 

Pure  mica 0 . 0091 

Built-up  mica 0.0031  to  0.0026 

Empire  cloth,  tightly  wrapped  (no 

air  spaces) 0 . 0063 

Presspahn 0.0042 

The  maximum  temperatures  to  which  insulating  materials 
may  be  subjected  should  not  exceed  the  following  limits: 

Asbestos 500°C.  or  more 

Mica  (pure) 500°C.  or  more 

Micanite. 125°  to  130°C. 

Presspahn,    leatheroid,    empire    cloth, 
cotton  covering,  insulating  tape,  and 

similar  materials 90°  to  95°C. 

If  ventilating  ducts  are  provided  at  sufficiently  frequent 
intervals  to  ensure  that  the  internal  temperatures  will  not  be 
greatly  in  excess  of  the  surface  temperatures,  it  is  merely  nec- 
essary to  see  that  the  cooling  surface  is  sufficient  to  dissipate 
the  watts  lost  in  the  iron  and  copper  of  the  armature. 

Temperature  Rise  of  Self-ventilating  Machines. — In  calculating 
the  losses  and  the  cooling  surfaces  of  the  armature,  we  shall 
assume  that  the  current  density  in  the  conductors  has  been 
so  chosen  that  the  end  connections  will  not  be  appreciably  hotter 
than  the  armature  as  a  whole.  If  this  density  does  not  exceed 
the  value  as  calculated  by  formula  (51)  of  Art.  29,  it  may  be 
assumed  that  the  temperature  rise  of  the  end  connections  will 


LOSSES  IN  ARMATURES  109 

not  exceed  40°C.,  and  the  watts  to  be  dissipated  by  the  cooling 
surfaces  of  the  armature  core  will  consist  of: 

1.  The  hysteresis  and  eddy-current  losses  in  the  teeth. 

2.  The  hysteresis  and  eddy-current  losses  in  the  core  below 
the  teeth. 

3.  The  PR  losses  in  the  " active"  portion  of  the  armature 
winding. 

All  these  losses  can  be  calculated  in  the  manner  previously 
explained.  The  copper  loss  to  be  taken  into  account  is  not 
the  total'/2/?  loss  in  the  armature  winding,  but  is  this  total  loss 

21 
multiplied  by  the  ratio  ^ —  [-y>  where  la  is  the  gross  length  of 

Zla  -\-  Le 

the  armature  core,  and  le  is  the  length  of  the  end  connections 
of  one  coil,  as  calculated  by  formula  (52)  of  Art.  29. 

The  various  cooling  surfaces  may  be  considered  separately, 
and  the  watts  carried  away  from  each  surface  computed 
independently.  The  total  cooling  surface  may  conveniently 
be  divided  into: 

1.  The  outside  cylindrical  surface  of  the  (revolving)  armature. 

2.  The  inside  cylindrical  surface  over  which  the  air  passes 
before  entering  the  radial  cooling  ducts. 

3.  The  entire  sur-face  of  the  radial   ventilating  ducts,   and 
the  two  ends  of  the  armature  core. 

The  cooling  effect  of  the  external  surface  at  the  two  ends  of 
the  armature  core  is  generally  similar  to  that  of  the  radial  venti- 
lating spaces,  and  it  is  convenient  to  think  of  the  two  end  rings 
as  being  equivalent  to  an  extra  duct.  Thus,  in  Fig.  35,  the 
number  of  ducts  is  shown  as  five,  and  the  cooling  surface  of  each 

duct   (both  sides)   is  ~  (D2  •-  d2).     The  calculations   would  be 

made  on  the  assumption  that  there  are  six  ducts.  If  the  number 
of  radial  vent  ducts  provided  is  n,  the  total  cooling  surface  of 
the  ducts  and  the  two  ends  of  the  armature  will  be 

I  (£>2  -  <*2)(n  +  1). 

The  outside  cylindrical  surface  of  the  armature  will  be  taken 
as  irDla,  where  la  is  the  gross  length,  no  deduction  being  made 
for  the  space  taken  up  by  the  vent  ducts.  The  cooling  surface 
of  the  end  connections  beyond  the  core  is  not  taken  into  account. 

The  area  of  the  inside  cylindrical  surface  is  irdla. 


110          PRINCIPLES  OF  ELECTRICAL  DESIGN 


/1,500+jA 
'  1A  \  100,000  /  (53) 


The  watts  dissipated  by  the  cylindrical  cooling  surfaces  may 
be  calculated  by  the  formula 

1,500  +  v\ 

100,000 
where   W  =  the  watts  dissipated. 

T  =  the  surface  temperature  rise  in  degrees  Centigrade. 
A  =  the  cooling  area  in  square  inches. 
v  =  the  peripheral  velocity  in  feet  per  minute. 

This  formula  is  generally  similar  to  one  proposed  some  years 
ago  by  DR.  GISBERT  KAPP. 

If  wc  is  a  cooling  coefficient  representing  the  watts  that  can 
be  dissipated  per  square  inch  of  surface  for  1°C.  difference  of 
temperature,  we  have 

1,500  +  v  ,'. 


and  the  temperature  rise  will  be 

r-  w 


wcA 

where  W  stands  for  the  watts  that  have  to  be  dissipated  through 
the  cooling  surface  A. 

The  watts  dissipated  by  the  air  ducts  and  end  surfaces  may 
be  calculated  by  the  formula 

w  -  TA  (55) 


where  W,  T,  and  A  have  the  same  meaning  as  before,  but  vd 
stands  for  the  average  velocity  of  the  air  through  the  ducts  in 
feet  per  minute.  This  velocity  is  very  difficult  to  estimate  in 
the  case  of  self-ventilating  machines,  but  the  constant  in  the 
formula  has  been  selected  to  give  good  average  results  if  v*  is 
taken  as  one-tenth  of  the  peripheral  velocity  of  the  armature. 

If  Wd  is  a  cooling  coefficient  representing  the  watts  that  can 
be  dissipated  per  square  inch  of  duct  surface  for  each  degree 
Centigrade  rise  of  temperature,  we  have 

/K«\ 


and  the  temperature  rise  of  the  vent  duct  surfaces  will  be 


where  W  stands  for  the  watts  that  have  to  be  dissipated  through 
the  surface  of  area  A. 


LOSSES  IN  ARMATURES 


111 


Example. — In  order  to  explain  the  application  of  the  formulas 
for  armature  heating,  numerical  values  will  be  assumed  for  the 
dimensions  in  Fig.  35. 
Let     D  =  32  in., 
d  =  23  in., 
la  =  15  in., 

n  =  5  (the  number  of  air  ducts  in  the  armature  core), 
N  =  400  revolutions  per  minute, 
whence 


and 


v  =  400  X  -^  =  3,360 


12 


vd  =  336. 


FIG.  35. — Section  through  armature  core. 

Let  us  further  assume  that  the  total  armature  losses,  con- 
sisting of  hysteresis  and  eddy-current  losses  in  teeth  and  core, 
together  with  the  PR  losses  in  the  portion  of  the  armature  wind- 
ing that  is  buried  in  the  slots,  amount  to  7  kw. 

The  cooling  surfaces  to  be  considered  are: 

1.  The  outside   cylindrical  surface    of  area  A\  =  TT  X  32  X 
15  =  1,510  sq.  in. 

2.  The   inside    cylindrical    surface    of    area   A*  =  TT  X  23  X 
15  =  1,085  sq.  in. 


112          PRINCIPLES  OF  ELECTRICAL  DESIGN 

3.  The  ventilating  duct  surface,  including  the  two  ends  of 

the  armature  coil,  of  area  A3  =  ^  (32*  -  232)  (  5+  1)  =  4,680 

z 

sq.  in. 

The  radiating  coefficients  to  be  used  are  calculated  by  formulas 
(54)  and  (56);  thus,  for  surface  (1), 

1,500  +  3,360 
Wc=         100,000 

and  the  watts  that  can  be  dissipated  per  degree  rise  of  temperature 
are 

Wi  =  wcAi 

=  0.0486  X  1,510  =  73.4 
For  surface  (2), 

1,500  +  (3,360  X  §) 

-  -  -  100,000  -  °'0391 

and  the  watts  that  can  be  dissipated  per  degree  rise  of  temperature 
are 

Wz  =  wcA2 

=  0.0391  X  1,085  =  42.5 
For  surface  (3), 

a°1344 


and  the  watts  that  can  be  dissipated  per  degree  rise  of  temperature 
are 

Ws  =  wdA, 

=  0.01344  X  4,680  =  63 

The  total  watts  that  can  be  dissipated  per  degree  rise  of  tem- 
perature are  73.4  +  42.5  +  63  =  178.9;  whence  the  rise  in 
temperature  to  be  expected  will  be 

7'000   _   oo  o°C 

17^9  - 

Temperature  Rise  of  Machines  with  Forced  Ventilation.  —  When 
a  machine  is  designed  for  forced  ventilation,  suitable  ducts  — 
whether  radial  or  axial  —  must  be  provided  in  the  armature,  and 
the  frame  must  be  so  arranged  as  to  provide  proper  passages  for 
the  incoming  and  outgoing  air.  The  fan  or  blower  may  be  out- 
side or  inside  the  enclosing  case.  It  is  usual  to  allow  100  cu.  ft. 
of  air  per  minute  for  every  Idlowatt  lost  in  heating  the  arma- 


LOSSES  IN  ARMATURES  113 

ture  and  field  coils  of  the  machine.  This  will  result  in  a  dif- 
ference of  about  20°C.  between  the  average  temperatures  of  the 
outgoing  and  incoming  air. 

In  order  to  ensure  that  there  shall  be  no  unduly  high  local 
temperatures  in  the  machine,  the  air  ducts  or  passages  must  be 
suitably  proportioned,  and  provided  at  frequent  intervals.  When 
the  paths  followed  by  the  air  through  the  machine,  and  the 
cross-section  of  these  air  channels,  are  known,  the  average  velocity 
of  the  air  over  the  heated  surfaces  can  be  calculated.  Formula 
(56)  can  then  be  used  for  determining  approximately  the  dif- 
ference in  temperature  between  the  cooling  surfaces  and  the 
air. 

35.  Summary,  and  Syllabus  of  Following  Chapters. — All 
necessary  particulars  have  been  given  in  this  and  the  fore- 
going chapters  for  determining  approximately  the  dimensions 
and  windings  of  an  armature  suitable  for  a  given  output  at  a 
given  speed.  A  suggested  method  of  procedure  in  deisgn 
will  be  explained  later;  but,  so  far  as  the  preliminary  design 
of  the  armature  is  concerned,  the  dimensions  are  determined 
by  using  an  output  formula  (Art.  19,  Chap.  IV)  and  deciding 
upon  the  diameter  D  and  the  length  la  of  the  armature  core. 
When  proportioning  the  slots  to  accommodate  the  winding,  the 
diameter  D  should  be  definitely  decided  upon,  but  slight  altera- 
tions in  the  length  la  can  readily  be  made  later  if  it  is  found 
necessary  to  modify  the  amount  of  the  flux  per  pole  (<l>)  or  the 
air-gap  density  (Bg).  Once  the  calculations  for  temperature  rise 
have  been  made,  and  the  design  so  modified — if  necessary— 
as  to  keep  this  within  40°  to  45°C.,  the  dimensions  of  the  arma- 
ture will  require  no  further  modification.  The  question  of 
commutator  heating  will  be  taken  up  later;  but,  in  designing 
the  armature  for  a  given  temperature  rise,  the  assumption  is 
made  that  no  appreciable  amount  of  heat  will  be  conducted  to 
or  from  the  commutator  through  the  copper  lugs  connecting  the 
armature  winding  to  the  commutator  bars. 

The  flux  per  pole  necessary  to  generate  the  required  voltage 
being  known,  the  remainder  of  the  problem  consists  in  designing 
a  field  system  of  electromagnets  capable  of  providing  the  re- 
quired flux  in  the  air  gap.  This  problem  is  similar  to  that  of 
designing  an  electromagnet  for  any  other  purpose,  and  it  has 
been  considered  in  some  detail  in  Chaps.  II  and  III.  It  might 
appear,  therefore,  that  little  more  need  be  said  in  connection  with 


114          PRINCIPLES  OF  ELECTRICAL  DESIGN 

the  design  of  a  continuous-current  generator,  but  it  must  be 
remembered  that  certain  assumptions  were  made  in  order  that 
the  broad  questions  of  design  might  not  be  obscured  by  too  much 
detail,  and  in  order  also  that  the  leading  dimensions  of  the 
machine  might  be  decided  upon. 

It  was  assumed  that  the  flux  in  the  air  gap  was  uniformly  dis- 
tributed under  the  pole  face;  but  is  it  so  distributed,  and  if  not, 
how  does  this  affect  the  tooth  saturation  and  the  ampere-turns 
required  to  overcome  the  reluctance  of  the  teeth  and  gap? 
What  is  the  influence  of  the  air-gap  flux  distribution  on  arma- 
ture reaction  and  voltage  regulation,  and  how  can  we  calculate 
the  field  excitation  required  at  different  loads  in  order  that  the 
proper  terminal  voltage  may  be  obtained?  These  and  similar 
questions  cannot  be  answered  without  a  more  thorough  study 
of  the  magnetic  field  cut  by  the  conductors,  at  full  load  as  well 
as  on  open  circuit. 

Again,  with  a  non-uniform  field  under  the  poles,  the  flux 
density  in  the  teeth  may  be  much  higher  than  would  be  indicated 
by  calculations  based  on  a  uniform  field,  and  this  might  lead  to 
excessive  heating. 

Perhaps  the  most  important  problem  in  the  design  of  direct- 
current  machines  is  that  of  commutation  which,  so  far,  has  barely 
been  touched  upon.  It  is  proposed  to  devote  a  whole  chapter 
to  the  study  of  commutation  phenomena. 

These  various  matters  will  be  taken  up  in  the  following 
order:  First,  a  study  of  the  flux  distribution  over  the  armature 
surface,  and  what  follows  therefrom  in  relation  to  tooth  densities, 
regulation,  and  the  excitation  required  at  various  loads;  next, 
commutation  and  the  design  of  commutating  poles;  and  finally, 
some  notes  on  the  design  of  the  field  system,  with  a  brief 
reference  to  the  factors  that  must  be  taken  into  account  when 
calculating  the  efficiency  of  a  continuous-current  generator. 


CHAPTER  VII 
FLUX  DISTRIBUTION  OVER  ARMATURE  SURFACE 

An  experienced  designer  may  go  far  and  obtain  good  results 
without  resorting  to  the  more  or  less  tedious  process  of  plotting 
flux-distribution  curves;  but  occasions  arise  when  his  experience 
and  judgment  fail  him,  and  when  a  reasonably  accurate  method 
of  predetermining  the  distribution  of  flux  density  over  the  sur- 
face of  the  armature  would  give  him  all  necessary  information. 
A  method  of  designing  electric  machinery — whether  continuous- 
current  dynamos  or  alternating-current  generators — which  in- 
volves the  plotting  of  the  flux  distribution  curves,  has  much  to 
recommend  it,  not  only  to  the  student,  but  also  to  the  professional 
designer.  The  advantage  from  the  studentrs  point  of  view  is 
that  a  more  accurate  conception  of  the  operating  conditions 
can  be  obtained  than  by  using  empirical  formulas,  or  making 
the  unscientific  assumptions  which  are  otherwise  necessary. 
The  designer  will  be  glad  to  avail  himself  of  a  practical  method 
of  plotting  flux  curves  when  departures  have  to  be  made  from 
standard  models,  or  when  it  is  desired  to  investigate  thoroughly 
the  effects  of  cross-magnetization  upon  commutation  or  pressure 
regulation. 

36.  Air-gap  Flux  Distribution  with  Toothed  Armatures.— 
The  determination  of  the  flux  densities  in  all  parts  of  the  tooth 
and  slot  for  various  values,  of  the  average  air-gap  flux  density 
is  so  difficult  and  complicated  that  it  is  safe  to  say  no  correct 
mathematical  solution  may  be  looked  for,  although  empirical 
rules  and  formulas  of  great  practical  value  may  serve  the 
purpose  of  the  designer. 

The  reluctance  of  the  magnetic  paths  between  pole  face  and 
armature  core  can  be  calculated  with  but  little  error  for  the  two 
extreme  cases  of  very  low  and  very  high  average  flux  density 
over  the  tooth  pitch;  but  for  intermediate  values  the  designer 
has  still  to  rely  on  his  judgment,  based  on  familiarity  with  the 
laws  of  the  magnetic  circuit. 

To  calculate  the  permeance  of  the  air  paths  over  one  slot 
pitch  at  the  center  of  the  pole  face,  when  the  density  is  low,  the 
magnetic  lines  are  supposed  to  follow  the  paths  indicated  in 

115 


116 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


Fig.  36.  The  tooth  is  drawn,  for  convenience,  with  parallel 
sides,  and  the  magnetic  lines  entering  the  sides  of  the  tooth  are 
supposed  to  follow  a  path  consisting  of  a  straight  portion  of 
length  5,  equal  to  the  actual  clearance,  and  a  circular  arc  of 
radius  r,  all  as  indicated  in  the  figure.  This  is  obviously  an 
arbitrary  assumption,  but  it  is  convenient  for  calculation  and 
gives  very  good  results.  It  agrees  very  closely  with  the  results 
obtained  by  MESSRS.  H.  S.  HELE-SHAW,  ALFRED  HAY,  and  P. 
H.  POWELL  in  their  classic  Institution  paper1  and  also  with  the 
i  i 

W&Z 


FIG.  36. — Flux  lines  entering  toothed  armature.     (Low  flux  density.) 

correct  mathematical  conclusions  arrived  at  by  MR.  F.  W. 
CARTER2  based  on  certain  assumptions,  including  that  of  infinite 
permeability  of  the  iron  in  the  teeth. 

Considering  a  portion  of  the  air  gap  1  cm.  long  axially  (i.e., 
in  a  direction  normal  to  the  plane  of  the  section  shown  in  Fig. 
36),  the  permeance  over  the  slot  pitch  of  width  X  is  seen  to  be 
made  up  of  two  parts:  (1)  the  permeance  PI  between  pole  face 

and  top  of  tooth,  of  value  PI  =  T,  and  (2)  the  permeance  2P2 

where  P2  is  the  permeance  between  the  pole  face  and  one  side 
of  the  tooth.  Consider  any  small  section  of  thickness  dr  as 
indicated  in  Fig.  36.  The  permeance  of  such  a  path,  of  depth 
1  cm.  measured  axially,  is 

dr 


dr 


2 

-  X 

7T 


1  "  Hydrodynamical  and  Electromagnetic  Investigations  Regarding  the 
Magnetic-flux  Distribution  in  Toothed-core  Armatures,"  Proc.  Inst.  E.  E., 
vol.  34,  p.  21. 

2  Electrical  World,  vol.  38,  Nov.  30,  1901,  p.  884. 


and 


FLUX  DISTRIBUTION 
2  r*     dr 


117 


The  average  permeance  per  square  centimeter  over  the  slot  pitch 
at  center  of  pole  is,  therefore, 

Pi  +  2P2 


sq.  cm. 


, 


(57) 


The  reciprocal  of  this  quantity  is  the  reluctance  per  square 
centimeter  of  cross-section,  or  the  equivalent  air-gap  length  5e. 
Thus 

t  +  S  (58)' 


Consider  now  Fig  37,  which  illustrates  the  case  of  a  highly 
saturated  tooth.     The  lines  of  flux  are  shown  parallel  over  the 


1  1  1  j  1  II  1  1  1  1 

3 

*"•«- 

. 

.^--t—  * 

t 

1 

I 

FIG.  37. — Flux  lines  entering  toothed  armature.     (High  flux  denxity.) 

whole  of  the  slot  pitch,  a  condition  which  is  approached — but 
never  attained — as  the  density  in  the  tooth  is  forced  up  to  higher 
and  higher  values.  It  is  obviously  only  when  the  permeability 

1  If  the  ventilating  ducts  are  closely  spaced,  or  exceptionally  wide,  the  gap, 
5t,  for  the  equivalent  smooth-core  armature,  as  given  by  formula  (58),  might 
have  to  be  slightly  modified;  but  the  calculation  of  fringing  at  the  sides  of 
vent  ducts  is  usually  an  unnecessary  refinement. 


118          PRINCIPLES  OF  ELECTRICAL  DESIGN 

of  the  iron  in  the  tooth  becomes  equal  to  unity  —  that  is  to  say, 
equal  to  the  permeability  of  the  air  paths  —  that  this  parallelism 
of  the  flux  lines  would  occur,  and  the  equivalent  air  gap  would 
be  be  =  5,  to  which  would  have  to  be  added  another  air  gap  of 
length  d  (Fig.  37)  to  represent  the  reluctance  of  the  teeth  and 
slots.  This  is  an  extreme,  and  indeed  an  impossible,  condition; 
but,  since  the  actual  distribution  of  the  lines  of  flux  in  tooth  and 
slot  cannot  be  predetermined,  the  calculations  for  very  high 
densities  are  usually  made  by  assuming  the  flux  lines  to  be 
parallel,  as  indicated  in  Fig.  37.  It  is  when  this  assumption  is 
made  for  low  values  of  the  density  that  appreciable  errors  are 
likely  to  be  introduced.  The  following  method  of  calculating 
the  joint  reluctance  of  tooth  and  slot  should  not  be  used  for  tooth 
densities  below  20,000  gausses. 

Considering  1  cm.  only  of  axial  net  length  of  armature  core 
(i.e.,  I  cm.  total  thickness  of  iron),  the  reluctance  of  the  air 
gap  and  tooth,  taken  over  the  width  of  one  tooth  only,  is, 

_  d       d  _  (d  +  Mg) 
Kl  ~  t  +  id  ~        tf 

The  reluctance  of  the  slot  portion  of  the  total  tooth  pitch  is, 


The  air  gap  of  the  equivalent  smooth-core  armature  —  being  the 
reluctance  per  square  centimeter  or  the  reciprocal  of  the  perme- 
ance per  square  centimeter  —  is,  therefore, 


e= 


which  can  be  put  in  the  form, 

(*+«)(<*+«) 


(59) 


This  equivalent  air  gap  includes  the  reluctance  of  the  tooth  itself 
when  the  flux  density  is  high,  but  does  not  take  account  of  the 
flux  in  the  vent  ducts  and  spaces  between  stampings.  It  is 
seen  to  depend  upon  the  permeability  of  the  iron,  and,  therefore, 
upon  the  actual  flux  density  in  the  tooth.  In  order  to  make  use 
of  formula  (59),  a  value  for  the  flux  density  in  the  tooth  must  be 
assumed.  A  method  of  working  which  involves  a  change  in  the 


FLUX  DISTRIBUTION  119 

equivalent  air  gap  for  various  values  of  the  flux  density  would 
be  unpractical,  and,  since  no  exact  method  is  ever  likely  to  be 
developed,  some  sort  of  compromise  must  be  made. 

Length  of  Air  Gap. — The  air-gap  clearance  5  must,  of  course, 
be  decided  upon  before  the  calculation  of  tooth  and  slot  reluctance 
can  be  made.  The  controlling  factor  in  determining  this  clear- 
ance is  the  armature  strength  or  the  ampere-turns  per  pole  of 
the  armature.  If  the  m.m.f.  due  to  the  armature  greatly  exceeds 
the  excitation  on  the  field  poles,  there  will  be  trouble  due  to  field 
distortion  under  load,  which  will  lead  to  poor  regulation  and 
commutation  difficulties.  The  field  ampere-turns  at  full  load 
should  be  greater  than  the  armature  ampere-turns.  A  safe  rule 
is  to  provide  an  air  gap  such  that  the  open-circuit  ampere-turns 
required  for  the  air  gap  alone — assuming  a  smooth  core  and  no 
added  reluctance  due  to  slots — would  be  equal  to  the  ampere- 
turns  on  the  armature  at  full  load.  Thus  if  (SI)g  are  the  ampere- 
turns  per  field  pole  required  to  overcome  the  reluctance  of  an 
air  gap  of  length  6,  we  may  write  (SI}0  =  (SI)a  where  (*$/)<, 
stands  for  the  armature  ampere-turns  as  calculated  by  formula 
(48)  page  80.  This  gives  for  6  the  value: 


or,  approximately, 


(SI)g  X  0.4rr  . 
2.54J9, 


where  Bg  may  be  taken  as  the  apparent  flux  density  in  the 
air  gap  under  the  pole  face  on  the  assumption  that  there  is  no 
fringing.  (For  approximate  values  of  Ba,  refer  to  the  table  on 
page  75.) 

The  length  of  air  gap  may  be  somewhat  reduced  if  corn- 
mutating  interpoles  are  provided,  especially  if  pole-face  windings 
(see  Art.  50,  Chap.  VIII)  are  used. 

Another  factor  which  may  influence  the  air-gap  clearance  is 
the  possibility  of  unbalanced  magnetic  pull  due  to  slight  decen- 
tralization of  the  armature.  This  becomes  of  importance  only 
in  machines  of  large  diameter  with  many  poles  and  rarely 
necessitates  a  clearance  greater  than  that  obtained  by  applying 
the  above  rule. 

37.  Actual  Tooth  Density  in  Terms  of  Air-gap  Density.— 
It  is  convenient  to  think  of  the  reluctance  of  air  gap,  teeth,  and 


120          PRINCIPLES  OF  ELECTRICAL  DESIGN 

slots  as  consisting  of  two  reluctances  in  series,  (a)  the  reluctance 
of  the  equivalent  air  gap  (as  calculated  by  formula  (58)  for  the 
center  of  the  pole  face),  and  (6)  the  reluctance  of  the  tooth. 
The  calculation  of  this  latter  quantity  depends  upon  a  knowledge 
of  the  actual  flux  density  in  the  tooth.  For  low  densities  in  the 
iron — up  to  about  20,000  gausses — the  actual  tooth  density  will 
be  approximately  equal  to  the  apparent  density;  that  is  to  say, 
practically  all  the  flux  entering  the  armature  over  one  tooth 
pitch  will  pass  into  the  core  through  the  root  of  the  tooth.  For 
densities  exceeding  20,000  gausses,  a  closer  estimate  of  the  correct 
value  of  the  tooth  density  may  be  made  by  assuming  the  con- 
dition of  Fig.  37. 
Let  the  meaning  of  the  symbols  be  as  follows: 

Bg  =  the  average  air-gap  flux  density  at  armature  surface; 
i.e.,  the  average  density  over  one  tooth  pitch  of  width 
t  -\-  s  and  length  la. 

Bt  =  the  actual  tooth  density. 

B8  =  the  density  in  the  slot  and  air  spaces. 

<I>x  =  the  total  flux  entering  armature  core  in  the  space 
of  one  slot  pitch. 

ln  =  the  net  length  of  the  armature  core  (iron  only). 
The  other  dimensions  as  given  on  the  sketch  Fig.  37. 

If  the  assumption  is  made  that  the  lines  of  flux  lie  in  a  plane 
exactly  perpendicular  to  the  axis  of  rotation,  it  might  be  argued 
that  the  flux  in  the  ventilating  ducts  and  in  the  insulating  spaces 
between  the  iron  laminations  does  not  enter  the  iron  of  the 
armature  core;  and  the  reluctance  of  the  paths  followed  by  this 
flux  would  therefore  be  very  high.  This  argument  is  not  justi- 
fied since  the  flux  lines  in  the  ventilating  ducts  will  actually 
find  their  way  into  the  core  immediately  below  the  bottom  of 
the  slots,  even  if  the  iron  in  the  teeth  is  practically  saturated. 
We  shall  therefore  assume  two  equipotential  surfaces,  one  being 
the  pole  face  and  the  other  being  the  cylindrical  surface  passing 
through  the  roots  of  the  teeth.  The  flux  density  in  the  air  ducts 
and  spaces  not  occupied  by  iron  will  therefore  be  the  same  as 
the  density,  Bs,  in  the  slots,  and  the  m.m.f  required  to  overcome 
the  reluctance  of  air  gap  proper  and  slot  will  be  the  same  as 
the  m.m.f.  required  to  overcome  the  reluctance  of  air  gap  proper 
and  tooth;  therefore 


FLUX  DISTRIBUTION  121 

whence 


R         r>  ,Arv. 

a  =  a  (rtjr+ij) 

Considering,  now,  the  total  flux  entering  the  armature  over 
one  slot  pitch,  this  is  made  of  two  parts: 

1.  The  flux  in  the  iron  of  the  teeth,  of  value  Bttln. 

2.  The  flux  in  the  slots  and  ducts,  of  value 

Bg    [Sla   +t(la~    ln)] 

or 

B,    (la\    -    lnt) 

The  total    flux  entering  through  one  slot    pitch  can  also  be 
expressed  in  terms  of  B0,  being: 

<J>x  =  Ba\la 
Thus 

Ba\la    =    Bttln   +   B,(\la    -    tln)  (61) 

Substituting  in  (61)  the  value  for  B,  given  by  formula  (60)  in 
terms  of  Bi}  and  solving  for  B0,  we  get: 


By  assuming  values  of  Bt  ranging  between  20,000  and  (say) 
26,000  gausses,  the  corresponding  values  of  B0  can  be  calculated 
by  formula  (62),  and  a  curve  plotted  from  which  values  of  Bt  can 
be  found  when  B0  is  known. 

The  fact  that  this  formula  is  based  on  assumptions  justified 
only  if  the  value  of  Bt  is  very  high  should  not  be  lost  sight  of. 
For  very  low  values  of  Bt  it,  may  be  assumed  that  all  the  flux 
entering  through  one  slot  pitch  passes  through  the  iron  of  the 
tooth.  This  leads  to  the  expression: 


B,  =  B.  (63) 

Curves  may  be  plotted  from  the  formulas  (62)  and  (63)  and  a 
working  curve,  which  shall  be  a  compromise  between  these  two 
extreme  conditions,  can  then  readily  be  drawn.  This  will  be 
done  when  working  out  a  practical  design  in  a  later  chapter. 

38.  Correction  for  Taper  of  Tooth.  —  The  assumption  of  parallel 
sides  to  the  tooth  is  justified  only  when  the  diameter  of  the 
armature  is  large  relatively  to  the  slot  pitch  or  when  taper  slots 
are  used  in  order  to  provide  a  uniform  cross-section  throughout 
the  whole  length  of  the  tooth.  The  dimension  t  in  formula  (62) 


122          PRINCIPLES  OF  ELECTRICAL  DESIGN 

should,  in  the  first  place,  be  the  width  of  the  narrowest  part  of 
the  tooth,  as  it  is  important  that  the  density  at  this  point  be 
known;  it  rarely  exceeds  25,000  gausses  in  continuous-current 
machines,  and  is  less  in  alternators.  When  the  field  system 
revolves,  as  in  most  modern  alternators,  the  armature  teeth 
will  usually  be  wider  at  the  root  than  at  the  top,  and  but  little 
error  will  be  introduced  by  taking  for  t  the  average  width,  for 
the  purpose  of  calculating  the  average  density  Bt  and  the  ampere- 
turns  required  for  the  teeth. 

The  case  of  a  tooth  with  considerable  taper,  in  which  the 
density  at  root  is  in  excess  of  10,000  gausses,  may  be  dealt  with 
by  the  application  of  SIMPSON'S  rule.  Having  determined  the 
density  Bt  at  the  root  of  the  tooth,  by  applying  formula  (62)  or 
(63)  as  the  case  may  demand,  the  assumption  is  then  made  that 
the  total  flux  in  the  tooth  remains  unaltered  through  other 
parallel  sections.1 


FIG.  38.  Taper  tooth. 

The  value  of  the  magnetizing  force  H  (or  the  ampere-turns 
required  per  unit  length)  can  then  be  determined  for  any  section 
of  the  tooth  by  referring  to  the  B-H  curves  for  the  iron  used  in 
the  armature.  It  is  sufficient  to  determine  H  for  three  sections 
only. 
Let  these  values  be: 

Hn  at  narrowest  section 
Hw  at  widest  section 

(7>          [        D 
i.e.,  where  the  value  of  Bm  is  -  ^— ~ — 

1  This  is  not  a  correct  assumption  when  the  root  density  is  very  high,  be- 
cause in  that  case  flux  will  leak  out  from  the  sides  of  the  tooth  to  the  bottom 
of  the  slot;  and  at  some  distance  from  the  bottom  of  slot  (the  taper  being 
as  indicated  in  Fig.  38)  the  total  flux  in  the  tooth  will  be  greater  than  at  the 
root  cross-section. 


FLUX  DISTRIBUTION 


123 


Then,  on  the  assumption  that  the  portion  of  the  B-H  curve 
involved  is  a  parabola,  SIMPSON'S  approximation  is, 

average  H  =  %Hn  +  %Hm  +  %HW  (64) 

Referring  to  Fig.  38,  it  will  be  seen  that  Hw  is  taken  at  the 
section  which  would  be  the  top  of  the  tooth  if  the  air  gap  were 
increased  from  5  to  the  " equivalent"  value  de  as  calculated  by 
formula  (58).  This  is  recommended  as  a  good  practical  com- 
promise; and  the  m.m.f.  in  gilberts  required  to  overcome  the 
reluctance  of  the  tooth  is  H  X  de  where  de,  the  equivalent  length 
of  tooth,  must  be  expressed  in  centimeters.  If  preferred,  the 
formula  (64)  can  be  modified  to  give  an  average  value  of  the 
necessary  ampere-turns  per  inch. 

39.  Variation  of  Permeance  over  Pole  Pitch — Permeance 
Curve. — The  permeance  per  square  centimeter  of  the  air  gap 
when  the  armature  is  slotted  may  be  calculated  for  the  center  of 
the  pole  face,  by  using  formula  (57).  This  value  will  not  change 
appreciably  for  other  points  under  the  pole  shoe  if  the  bore  of 
the  field  magnets  is  concentric  with  the  armature;  but  near  the 
pole  tips,  and  in  the  interpolar  space,  it  will  decrease  at  a  more 
or  less  rapid  rate,  depending  on  the  geometric  configuration  of 


A'*B  E     F 

FIG.  39. — Flux  lines  in  air  gap  of  dynamo.     (One  pole  acting  alone.) 

the  pole  pieces,  and  their  circumferential  width  relatively  to  pole 
pitch  and  air-gap  length.  In  considering  the  reluctance  of  the 
air  paths  between  pole  shoe  and  armature,  it  is  convenient  to 
think  of  an  equivalent  air  gap  of  length  8e  as  calculated  by  formula 
(58)  of  Art.  36;  and  in  the  following  investigation  the  actual 
toothed  armature  must  be  thought  of  as  being  replaced  by 
an  imaginary  smooth-core  armature  of  the  proper  diameter 
to  insure  that  the  reluctance  of  the  air  gap  per  unit  area  at  any 


124 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


point  on  the  periphery  shall  be  the  same  as  the  average  reluctance 
per  unit  area  taken  over  the  slot  pitch. 

In  Figs.  39  and  40  an  attempt  has  been  made  to  represent 
the  actual  distribution  of  flux  lines  (1)  for  the  condition  of  one 
pole  acting  alone  without  interference  from  neighboring  poles, 
and  (2)  for  the  practical  condition  of  neighboring  poles  of  equal 
strength  and  opposite  polarity.  The  machine  to  which  these 
diagrams  apply  is  a  continuous-current  dynamo  of  pole  pitch 
37  cm.,  pole  arc  27  cm.,  and  equivalent  air  gap  of  0.8  cm.  at 
center  of  pole  face.  The  air  gap  is  of  uniform  length  except 
near  the  pole  tips,  where  it  is  slightly  increased,  as  indicated 


A  N 

FIG.  40. — Flux  lines  in  air  gap  of  dynamo.     (Effect  of  neighboring  poles.) 

on  the  drawings.  With  a  little  practice,  unlimited  patience,  and 
ample  time  in  which  to  perform  the  work,  diagrams  of  flux  dis- 
tribution such  as  those  of  Figs.  39  and  40  can  be  drawn,  and 
they  will  indicate  accurately  the  actual  arrangement  of  the 
flux  lines.  The  method  is  one  of  trial  and  gradual  elimination 
of  errors,  based  on  the  well-known  principle  that  the  space 
distribution  of  the  flux  lines  will  be  such  as  to  correspond  with 
maximum  total  permeance,  or,  in  other  words,  such  as  will 
produce  the  maximum  flux  with  a  given  m.m.f. 

Probably  one  of  the  most  practical  and  at  the  same  time 
most  accurate  methods  of  procedure  is  that  proposed  by  DR. 
LEHMANN1  and  followed  in  preparing  the  flux  diagrams  (Figs. 
39,  40,  42  and  43) .  A  section  perpendicular  to  the  shaft  through 
the  pole  shoe  and  armature  is  considered,  and  all  flux  lines  in 

1  "Graphische  Methode  zur  Bestimmung  des  Kraftlinienverlaufes  in  der 
Luft,"  Elektrotechnische  Zeitschrift,  vol.  30  (1909),  p.  995. 


FLUX  DISTRIBUTION  125 

the  air  gap  are  supposed  to  lie  in  planes  parallel  to  this  section. 
Equipotential  lines  are  drawn  in  directions  which  seem  reason- 
able to  the  draughtsman,  and  tubes  of  flux,  all  having  the  same 
permeance,  are  then  drawn  with  their  boundary  lines  perpen- 
dicular at  all  points  to  the  equipotential  lines.  At  the  first  trial 
it  will  generally  be  found  that  these  conditions  cannot  be  fulfilled, 
but  by  altering  the  direction  of  the  tentative  equipotential 
lines  the  work  is  repeated  until  the  correct  arrangement  of 
lines  is  obtained.  The  tube  of  induction  A  BCD  (Fig.  39)  is  the 
first  to  be  drawn.  Its  permeance  in  the  particular  case  con- 
sidered is  0.25  because  it  consists  of  four  portions  in  series,  each 
one  of  which  is  exactly  as  wide  as  it  is  long  (a  thickness  of  1  cm. 
measured  axially  is  assumed).  Proceeding  outward  from  left 
to  right,  and  making  each  section  of  the  individual  tube  of 
induction  exactly  as  wide  as  it  is  long,  the  permeance  of  every 
one  of  the  component  areas  in  the  diagram  is  always  unity,  and 
any  complete  tube,  such  as  EFGH,  has  the  same  permeance  (in 
this  example  0.25)  as  every  other  tube.  The  computation  of 
the  total  permeance  between  the  pole  shoe  and  armature  over 
any  given  area  is  thus  rendered  exceedingly  simple.  Although 
the  armature  surface  is  represented  as  a  straight  line  in  the 
accompanying  illustrations,  the  actual  curvature  of  the  armature 
may  be  taken  into  account  if  preferred;  but  the  error  introduced 
by  substituting  the  developed  armature  surface  for  the  actual 
circle  is  generally  negligible. 

In  Fig.  40  the  flux  lines  have  been  drawn  to  ascertain  the 
effect  of  the  neighboring  pole  in  altering  the  distribution  over 
the  armature  surface  in  the  interpolar  space.  The  perpendicular 
AW  has  been  erected  at  the  geometric  neutral  point,  and  may 
be  considered  as  the  surface  of  an  iron  plate  forming  a  con- 
tinuation of  the  armature  surface  AN.  Thus  ANN'  will  be  an 
equipotential  surface  between  which  and  the  polar  surface 
the  intermediate  equipotential  surfaces  must  lie. 

It  may  be  mentioned  tha,t  in  Figs.  39  and  40,  and  also  in 
the  other  flux-line  diagrams,  the  pole  core  under  the  windings 
cannot  properly  be  considered  as  being  at  the  same  magnetic 
potential  as  the  pole  shoe,  relatively  to  the  armature.  The 
proper  correction  can  be  introduced  in  calculating  the  flux  in 
each  tube  of  induction;  but  since  the  present  investigation  is 
confined  to  the  flux  entering  the  armature  from  the  pole  shoe,  it 
will  not  be  necessary  to  make  this  correction. 


126 


PRINCIPLES  OF  ELECTRICAL  DESIGN  . 


The  flux  density  at  all  points  on  the  armature  periphery  is 
easily  calculated  when  the  flux  lines  have  been  drawn.  Thus, 
since  each  tube  of  induction  encloses  the  same  number  of  mag- 
netic lines,  exactly  the  same  amount  of  flux  will  enter  the  arma- 
ture in  the  space  EF  (Fig.  39)  as  in  the  space  A B.  If  Bab  is 
the  flux  density  in  the  tube  CDAB  at  the  center  of  the  pole  face, 
the  average  density  over  the  space  EF  will  be 

Bef    =   Bab    X   Wp 

Thus  curves  of  flux  distribution  such  as  Fig.  41  can  readily 
be  drawn.  It  will  be  seen  that  the  dotted  curve,  giving  actual 
distribution  of  flux  for  the  case  of  Fig.  40,  does  not  differ  from 


Flux  Distribution, 
One  Pole  only 


O  Surf  ace  of  Armature  Core  N 

FIG.  41. — Curve  of  flux  distribution  over  armature  surface. 

the  full-line  curve  (case  of  Fig.  39,  with  no  interference  from 
neighboring  poles)  except  in  the  interpolar.  space  where  the  de- 
magnetizing effect  of  the  opposite  polarity  is  appreciable,  and 
causes  the  flux  to  diminish  rapidly  until  it  reaches  zero  value  on 
the  geometric  neutral  (the  point  N),  where  its  direction  re- 
verses. This  is  what  one  would  expect  to  find,  because,  although 
the  magnetic  action  of  any  one  pole  considered  alone  will  ex- 
tend far  beyond  each  pole  tip,  this  action  will  not  be  appreciable 
beyond  the  interpolar  space,  on  account  of  the  shading  effect 
of  the  neighboring  poles.  In  order  to  ascertain  how  far  the 
demagnetizing  effect  of  neighboring  poles  is  likely  to  extend 
when  the  air  gap  is  not  constant  but  increases  appreciably  in 


FLUX  DISTRIBUTION 


127 


length  as  the  distance  from  the  center  of  the  pole  increases,  the 
case  of  a  salient  pole  alternator  has  been  considered.  The  flux 
lines  and  equipotential  surfaces  for  an  alternator  with  shaped 
poles  are  shown  in  Figs,  42  and  43.  The  object  of  shaping  the 
poles  by  gradually  increasing  the  air  gap  from  the  center  out- 
ward is  to  obtain  over  the  pole  pitch  a  distribution  of  flux  which 


FIG.  42. — Flux  lines  in  air  gap  of  alternator.     (One  pole  acting  alone.) 


FIG.  43. — Flux  lines  in  air  gap  of  alternator.     (Effect  of  neighboring  poles.) 

shall  approximate  to  a  sine  curve.     The  data  of  the  machine 
under  consideration  are  as  follows: 

Pole  pitch  =  22  cm. 

Pole  arc  =  14.3  cm. 

Equivalent  air  gap  at  center  of  pole  face  5«  =  1  cm. 

Air  gap  at  other  points  on  armature  surface  =  - 

COS  v 


where 


0  is  the  angle  (in  electrical  degrees)  between  the  center  of  the 
pole  and  the  point  considered. 

In  Fig.  44  the  curve  marked  "permeance"  has  been  plotted 
from  Fig.  42.  Its  shape  indicates  the  flux  distribution  over  the 
armature  surface  on  the  assumption  that  the  effect  of  neighbor- 


128 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


ing  poles  is  negligible.  The  m.m.f.  between  pole  shoe  and  arma- 
ture core  being  the  same  at  all  points  on  the  armature  surface, 
it  is  evident  that  this  curve  of  flux  distribution  will  correctly 
represent  the  variations  of  air-gap  permeance  per  unit  area  of 
the  armature  surface.  Thus,  the  permeance  per  square  centi- 
meter at  the  point  7  cm.  from  center  of  pole  is  the  permeance 
of  the  tube  GHEF  in  Fig.  42  divided  by  the  area  of  the  surface 
EF.  The  permeance  of  the  tube  GHEF  is  exactly  the  same 
as  that  of  the  tube  CDAB,  i.e.,  0.25  per  centimeter  of  depth 
measured  axially.  The  area  of  the  surface  EF  is  0.5,  and  the 
permeance  per  square  centimeter  at  the  point  considered  is, 
therefore, 

*  -  is  -  «• 

40.  Open-circuit  Flux  Distribution  and  M.m.f.  Curves. — The 
dotted  curve  marked  "flux"  in  Fig.  44  has  been  plotted  from 


0     1    2    3   4     5    6    7    8    9  10  11  12  13 
Surface  of  Armature  Core 

FIG.  44. — Curves  of  permeance,  flux,  and  m.m.f. 

Fig.  43,  and  shows  the  effect  of  the  neighboring  pole  in  reducing 
the  air  gap  flux  and  causing  it  to  pass  through  zero  value  at  a 
point  exactly  halfway  between  the  poles.  A  comparison  of  the 
full  line  and  dotted  flux  curves  shows  that,  even  with  the  greatly 
increased  air  gap,  the  influence  of  the  neighboring  poles  is  not 
appreciable  except  in  the  uncovered  spaces  between  the  pole 
tips.  The  vertical  dotted  line  in  Fig.  44  shows  the  limit  of  the 
pole  arc. 

In  order  to  find  a  scale  for  the  flux  curve  it  is  necessary  to 
know  either  the  total  flux  entering  the  armature  in  the  space 


FLUX  DISTRIBUTION 


129 


of  a  pole  pitch  or  the  m.m.f.  between  pole  shoe  and  armature. 
If  the  resultant  m.m.f.  tending  to  send  flux  from  the  pole  to  any 
point  on  the  armature  is  known,  the  flux  density  can  be  calculated 
because,  B  =  flux  per  square  centimeter  =  (m.m.f.)  X  permeance 
per  square  centimeter. 

Thus,  if  the  m.m.f.  necessary  to  overcome  air-gap  reluctance 
at  center  of  the  pole  is  known,  the  curve  of  resultant  m.m.f. 
at  all  points  on  the  armature  can  be  plotted.  This  has  been 

done  in  Fig.  44,  where  the  ordinate  of  the  m.m.f.  curve  at  any 

i  n  —  R 

point,  such  as  10,  is  simply  10  —  M  --  —p>  where  the  or- 
dinate 10  —  B  of  the  flux  curve  must  be  expressed  in  gausses. 

41.  Practical  Method  of  Predetermining  Flux  Distribution. — 
Although  the  method  outlined  above,  gives  excellent  results 
in  the  hands  of  an  experienced  designer  who  can  afford  the  time 
required  to  map  out  the  actual  paths  of  the  flux  lines,  it  is  not 
suitable  for  general  use  in  actual  designing  work.  By  adopting 
a  simple  construction  which  assumes  a  certain  flux  distribution 
and  avoids  the  drawing  of  equipotential  lines,  results  of  sufficient 
accuracy  for  practical  purposes  can  very  quickly  be  obtained. 


A  BC      ODEF        G      H 

FIG.  45. — Approximate  flux  paths  between  pole  and  armature. 

This  construction  is  indicated  in  Fig.  45.  A  section  through 
one-half  of  the  pole  shoe,  showing  the  air  gap  in  its  proper  pro- 
portion, is  drawn  to  a  sufficiently  large  scale,  preferably  full 
size.  The  " developed"  armature  surface  may  be  used,  in 
which  case  the  construction  is  a  little  simpler  because  radial 
lines  may  be  shown  as  perpendiculars  erected  on  the  horizontal 
datum  line  representing  the  armature  surface.  The  distance 
AM  is  the  " equivalent"  air  gap,  as  calculated  for  the  center 
of  the  pole  face  by  using  formula  (58).  Draw  the  perpendicular 
OR  tangent  to  the  pole  tip,  and,  through  the  first  poiht  of 
tangency  Q,  draw  the  semicircle  BQF  with  its  center  at  0. 
Bisect  QO  at  the  point  P  and  draw  PD  at  an  angle  of  30  degrees 

9 


130 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


with  OQ .  Produce  DP  to  0'  where  it  meets  the  perpendicular 
erected  on  AD  at  the  point  B.  Flux  lines  of  which  the  length 
and  direction  will  be  approximately  correct  can  now  be  drawn. 
All  the  lines  from  points  on  armature  surface  lying  between  A 
and  B  will  be  considered  perpendicular  to  the  armature  surface, 
i.e.,  verticals  erected  on  the  datum  line.  Between  the  points 
B  and  D  the  lines  will  be  considered  straight,  but  with  a  slope 
determined  by  the  position  of  the  point  0'  through  which  they 
will  pass  if  produced  beyond  the  pole  face.  From  the  point  F 
the  equivalent  flux  line  will  be  the  arc  of  the  circle  described 
through  the  point  Q  with  the  radius  OQ,  and  lines  from  points 
between  D  and  F  must  be  put  in  by  the  eye  so  that  their  curva- 
ture shall  be  something  between  the  circle  through  F  and  the 
straight  line  through  D.  One  of  these  intermediate  lines  has 
been  drawn  from  the  point  E.  Over  the  region  beyond  F  all  flux 


FIG.  46. — Effect  of  neighboring  pole  in  modifying  flux  distribution. 

lines  will  be  drawn  as  circles  described  from  the  center  0  and 
continued  beyond  the  vertical  OR  until  they  meet  the  pole  in  a 
direction  normal  to  the  surface  of  the  iron.  Thus  the  line  from 
the  point  G  is  completed  by  an  arc  of  circle  with  its  center  at 
the  junction  of  the  line  OR  and  the  flat  surface  of  the  polar 
projection,  while  the  flux  line  from  H.  is  continued  as  a  straight 
line  (the  shortest  distance)  until  it  meets  the  pole  perpendicularly 
to  the  surface. 

Any  desired  number  of  lines  can  very  quickly  be  drawn  in 
this  manner,  and  they  may  be  thought  of  as  the  center  lines  of 
" equivalent"  tubes  of  flux  of  uniform  cross-section  over  their 
entire  length.  If,  now,  the  length  of  any  one  of  these  imaginary' 
flux  lines  be  measured  in  centimeters,  the  reciprocal  of  this 
length  will  be  the  permeance  per  square  centimeter  between  pole 
shoe  and  armature  at  the  point  considered.  It  is,  therefore, 
an  easy  matter  to  plot  a  permeance  curve  similar  to  the  one 


FLUX  DISTRIBUTION 


131 


shown  in  Fig.  44.  This  curve,  which  represents  the  permeance 
per  square  centimeter  of  armature  surface  between  pole  and 
armature,  can  evidently  be  thought  of  as  a  flux-distribution  curve 
on  the  assumption  that  one  pole  acts  alone  without  interference 
from  neighboring  poles. 

In  regard  to  the  actual  flux  distribution  for  no-load  conditions, 
it  may  be  argued  that  if  two  neighboring  poles  each  acting  alone 
would  produce  a  flux  distribution  as  shown  respectively  by  the 
full-line  and  dotted  curves  of  Fig.  46,  then  the  flux  at  any  point 
p  will  be  pm  —  pn.  This  method  of  plotting  the  resulting  flux- 
distribution  curve  should  give  satisfactory  results  in  the  space 


/ 

\ 

> 

/ 

| 
Eg  (Average) 

V 

VVlCurve  j 

*£ 

/ 

Curve    B       N^K^^ 

X 

• 

1 

A 

s 

^                                                                         T»_t_  T» 

\N 

:»_u    T                                                    ^J 

FIG.  47. — Practical  construction  for  deriving  flux  curve  from  permeance 

curve. 

between  pole  tips,  but  it  does  not  provide  for  the  gradual  change 
in  the  flux  distribution  near  the  pole  tips  where  the  shading  effect 
of  the  masses  of  iron  becomes  important.  For  the  practical 
designer  the  writer  recommends  the  approximation  indicated  in 
Fig.  47  where  P  is  the  permeance  curve  previously  obtained. 
The  flux  curve  is  derived  therefrom  by  drawing  the  straight 
line  RS,  connecting  the  point  on  the  permeance  curve  directly 
over  the  geometric  neutral  to  the  point  S  immediately  under  the 
pole  tip.  By  subtracting  from  the  ordinates  of  the  curve  P 
the  corresponding  ordinates  of  the  triangle  ORS,  the  curve  OA'N 
is  obtained,  representing  the  flux  distribution  on  open  circuit. 
This  curve  has  yet  to  be  calibrated,  because  the  value  of  its 
ordinates  cannot  be  determined  unless  either  the  m.m.f.  or  the 
total  flux  per  pole  is  known.  In  designing  a  machine,  the  total 


132          PRINCIPLES  OF  ELECTRICAL  DESIGN 

flux  per  pole  will  be  known  at  this  stage  of  the  work,  and  the 
unknown  factor  will  be  the  ampere-turns  on  the  poles  necessary 
to  produce  this  flux.  Measure  the  area  of  the  curve  OA'N  and 
construct  the  rectangle  OO'N'N  of  exactly  the  same  area.  The 
height  of  this  rectangle  will  be  a  measure  of  the  average  density 
over  the  pole  pitch.  This  is  known  to  be 

<£ 

Bg  (average)  = 


where  3>  =  total  flux  per  pole  in  the  air  gap. 
T  =  pole  pitch  in  centimeters. 
la  =  gross  length  of  armature  in  centimeters. 

In  this  manner  a  scale  is  provided  for  the  flux  curve  OA'N, 
which  should  preferably  be  replotted.  The  curve  of  resultant 
m.m.f.  over  armature  surface  can  now  be  derived  as  explained 
in  connection  with  Fig.  44. 

Since  the  permeance  curve  as  obtained  by  either  of  the  methods 
here  explained  does  not  take  into  account  the  reluctance  of  the 
armature  teeth,  or  indeed  the  reluctance  of  any  part  of  the 
magnetic  circuit  other  than  the  air  gap,  the  actual  ampere-turns 
necessary  to  produce  the  required  flux  will  be  greater  than  the 
amount  indicated  by  the  maximum  ordinate  of  the  m.m.f.  curve 
of  Fig.  44.  The  fact  that  the  reluctance  of  the  teeth  at  different 
points  under  the  pole  face  is  dependent  upon  the  flux  distribu- 
tion tends  to  complicate  the  problem,  but  a  method  of  accounting 
for  this  variation  will  be  explained  in  the  following  article. 

42.  Open-circuit  Flux-distribution  Curves,  as  Influenced  by 
Tooth  Saturation.  —  Before  considering  the  effect  of  the  armature 
current  in  altering  the  distribution  -of  magnetizing  force  over  the 
armature  periphery,  it  will  be  necessary  to  examine  briefly  how 
the  degree  of  saturation  of  the  teeth  may  be  taken  into  account 
and  a  correct  flux-distribution  curve  plotted.  The  method  about 
to  be  explained  is  due  to  PROFESSOR  C.  R.  MOORE,  it  is  probably 
more  easily  applied  and  less  tedious  than  an  equally  scientific 
method  more  recently  proposed  by  DR.  ALFRED  HAY.1 

In  Fig.  48  a  permeance  curve  has  been  drawn.  It  has  the 
same  meaning  as  the  curve  marked  "  Permeance"  in  Fig.  44 
(Art.  39),  and  it  may  be  obtained  for  any  given  machine  in  the 
manner  described  in  Art.  41.  If  this  curve  (Fig.  48)  has  been 

1  A.  HAY:  "Predetermining  Field  Distortion  in  Continuous-current 
Generators,"  Electrician,  72,  pp.  283-285,  Nov.  21,  1913. 


FLUX  DISTRIBUTION 


133 


calibrated  to  read  air-gap  permeance  per  square  centimeter  of 
armature  surface,  it  follows  that,  for  a  given  value  of  m.m.f. 
between  pole  and  armature,  the  flux  density  at  any  point — 
as  for  instance  d — will  be 

Bd  =  (m.m.f.)d  X  value  of  ordinate  of  Fig.  48  at  d, 
but  in  order  to  get  the  flux  into  the  armature  core  the  reluctance 
of  the  teeth  must  be  considered. 


FIG.  48. — Permeance  curve. 


O  Pb          Pa        Pe 

Ampere  -Turns  Required  for  Air-Gap,  Teeth  and  Slots 

FIG.  49. — Saturation  curves  for  air  gap,  teeth,  and  slots. 

For  any  value  of  the  air-gap  density  Bg  there  is  a  correspond- 
ing value  of  the  tooth  density,  Bt,  which  can  be  calculated  by 
formula  (62)  or  (63),  as  the  case  demands;  and  the  ampere-turns 
required  to  overcome  the  reluctance  of  the  tooth  can  be  found,  all 
as  explained  in  Art.  38.  This  value  can  be  plotted  in  Fig.  49 


134 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


against  the  corresponding  values  of  Bg,  and  the  resulting  curve 
shows  the  excitation  required  to  overcome  tooth  reluctance  for 
all  values  of  the  air-gap  density.  The  curves  for  the  air  gap 
proper  will  all  be  straight  lines  when  plotted  in  Fig.  49.  Let 
OR  be  the  curve  for  the  point  a  at  center  of  pole.  Add  the 
ampere-turns  required  for  the  teeth,  and  obtain  curve  (a), 
which  gives  directly  the  ampere-turns  required  to  overcome 
reluctance  of  air  gap,  teeth,  and  slots,  for  all  values  of  the  air- 
gap  density.  It  will  be  understood  that  the  ordinates  represent 
the  average  value  of  the  air-gap  density  at  armature  surface 


\a  b    c    d     e 
FIG.  50. — Open-circuit  m.m.f.  curve. 


\ 


over  a  slot  pitch.  Any  other  curve,  such  as  (d),  is  obtained 
by  first  drawing  a  straight  line  OQ  such  that  PQ  bears  to  PR 
the  same  relation  as  the  ordinate  at  d  in  Fig.  48  bears  to  the 
ordinate  at  a,  and  then  adding  thereto  the  ampere-turns  for  the 
teeth,  as  already  obtained.  The  curves  of  Fig.  49  should  in- 
clude a  sufficient  number  of  points  on  the  armature  surface;  and 
when  the  resultant  m.m.f.  between  armature  points  and  pole  is  also 
known,  the  correct  flux-distribution  curves  can  readily  be  plotted. 
Let  the  curve  of  Fig.  50  represent  the  distribution  of  the  re- 
sultant field  m.m.f.  on  open  circuit  obtained  as  explained  in  Art. 
40  (Fig.  44) ;  then,  at  any  point  such  as  e,  the  m.m.f.  is  given  by 
the  length  of  the  ordinate  ee'.  Find  this  value  on  the  horizontal 
scale  of  Fig.  49,  and  the  height  of  the  ordinate  at  this  point, 
where  it  meets  curve  e  (which  is  not  drawn  in  Fig.  49),  is  the 
flux  density,  which  can  be  plotted  as  ee'  in  Fig.  51.  In  this 
manner  the  flux  curve  A  of  Fig.  51  is  obtained.  The  area 
of  this  curve,  taken  between  any  given  points  on  the  armature 
periphery,  is  obviously  a  measure  of  the  total  flux  entering  the 
armature  between  those  points.  The  required  flux  per  pole  is 


FLUX  DISTRIBUTION 


135 


usually  known.  The  average  density  over  the  pole  pitch,  r, 
as  explained  in  Art.  41,  is 

3> 

Bg  (average)  =  ^r 

where  la  is  the  gross  length  of  armature.1  By  drawing  the  dotted 
rectangle  of  height  Bg(  average)  as  shown  in  Fig.  51,  its  area  can 
be  compared  with  that  of  curve  A  by  measuring  with  a  planim- 
eter.  If  these  areas  are  not  equal,  the  ampere-turns  per  field 
pole,  as  represented  by  the  curve  of  Fig.  50,  must  be  altered 
and  a  new  flux  curve  plotted,  of  which  the  area  must  indicate  the 
required  flux. 


FIG.  51. — Open-circuit  flux  distribution. 

43.  Effect  of  Armature  Current  in  Modifying  Flux  Distribution. 

—The  distribution  of  m.m.f.  over  armature  surface  when  the 

field  poles  are  acting  alone  has  already  been  calculated  and 

plotted  in  Fig.  50,  and  the  effect  of  the  armature  current  in 

modifying  this  distribution  may  be  ascertained  by  noting  that 

the  armature  ampere-turns  between  any  two  points  such  as  d 

and  d'  on  the  armature  circumference  (see  Fig.  52)  are  q(b  —  a), 

where  q  stands  for  the  specific  loading  or  the  ampere-conductors 

per  unit  length  of  armature  circumference. 

Let  p  =  number  of  poles, 

Z  =  total  number  of  conductors  on  armature  surface, 
Ic  =  current  in  each  conductor, 
T  =  pole  pitch, 

ZIe 

then    q  =  - 
pr 

1  The  gross  length  la  of  the  armature  core  usually  exceeds  the  axial  length 
of  the  pole  shoe  by  an  amount  equal  to  twice  the  air  gap,  5.  Thus,  by  as- 
suming the  flux  distribution  as  given  by  curve  A  of  Fig.  51  to  extend  to  the 
extreme  ends  of  the  armature,  a  practical  allowance  is  made  for  the  fringe 
of  flux  which  enters  the  corners  and  ends  of  the  armature  core. 


136 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


and  the  armature  m.m.f.  tending  to  modify  the  flux  due  to  the 
field  poles  alone  is 

ZIe(b  -  a) 


0.4 


pr 


between  the  points  d  and  d'. 

In  order  that  a  curve  may  be  plotted  on  the  same  basis  as  the 
resultant  field  m.m.f.  curve,  it  is  necessary  to  know  the  armature 
m.m.f.  per  pole  at  all  points.  Let  the  distance  between  the 
points  d  and  d'  be  equal  to  the  pitch  r\  then  the  effect  of  the  arma- 
ture m.m.f.  at  df  upon  the  pole  S  will  be  exactly  the  same  as 


S 

N 

S 

I 

^  d'  \£- 

^   d    \^ 

FIG.  52.  —  Magnetizing  effect  of  armature  inductors. 

the  effect  of  the  armature  m.m.f.  at  d  upon  the  pole  N,  and 
the  m.m.f.  per  pole  at  the  point  d  may  be  expressed  as 

0.47rZJc(b  -a) 


(Armature  m.m.f.)^  = 


(65) 


Its  maximum  value  —  which  always  occurs  in  the  zone  of  com- 
mutation —  is 


Armature  m.m.f.  per  pole  = 


(66) 


The  combination  of  this  armature  m.m.f.  with  the  m.m.f.  due 
to  the  field  coils  only,  as  represented  by  Fig.  50,  is  carried  out 
graphically  in  Fig.  53.  The  curves  F  and  A  represent  field  and 
armature  ampere-turns  (or  m.m.f.  in  gilberts,  if  preferred). 
These  are  the  two  components  of  a  resultant  m.m.f.  curve,  R, 
the  ordinates  of  which  are  a  measure  of  the  tendency  to  send  flux 
between  any  point  on  the  armature  surface  and  the  pole  shoe  N. 
The  actual  value  of  the  flux  density  at  the  various  armature 
points  under  load  conditions  can  therefore  be  obtained  by  using 


FLUX  DISTRIBUTION 


137 


the  curves  of  Fig.  49  and  plotting  the  values  of  air-gap  density 
corresponding  to  the  ampere-turns  read  off  the  curve  R  of  Fig.  53. 
The  procedure  is  exactly  the  same  as  when  obtaining  the  open- 
circuit  flux  curve  (Fig.  51)  by  using  the  values  of  Figs.  49  and 
50,  but  the  new  curve  of  flux  distribution — which  may  be  called 
curve  B,  to  distinguish  it  from  the  open-circuit  flux  curve  A— 
gives  the  distribution  over  armature  surface  for  a  given  brush 
position  and  a  specified  output.  The  difference  in  area  of 
curves  A  and  B  is  a  measure  of  the  flux  lost  through  armature 
reaction;  it  includes  not  only  direct  demagnetization,  but  also 
cross-magnetization  which — by  producing  distortion  of  the  flux 

Max.Field  S7  per  Pole 

'Resultant  M.M.F.under 
Load  ConditoDB 


Max.  Armature  57 
per  Pole 

I 

sh  Shift 

FIG.  53. — Addition  of  field  and  armature  m.m.fs. 

distribution — leads  sometimes  to  local  concentration  of  high 
densities  and  reduction  of  total  flux  owing  to  saturation  of|  the 
iron  in  the  armature  teeth. 

The  final  flux  curve  for  the  loaded  machine  is  generally  similar 
to  curve  B,  except  that  its  area  must  be  such  as  to  indicate  that 
the  desired  voltage  will  be  generated.  This  increased  area  is,  of 
course,  obtained  by  increasing  the  field  ampere-turns.  In  other 
words,  the  curve  F  of  Fig.  53  has  to  be  replaced  by  a  new  open- 
circuit  m.m.f.  curve  such  that  the  new  R  curve  resulting  from  its 
combination  with  the  existing  curve  A  will  produce  a  new  flux  curve 
similar  to  B,  but  of  the  required  area.  This  new  flux-distribution 
curve  may  be  called  curve  C,  to  distinguish  it  from  the  open- 
circuit  curve  A  and  also  from  the  flux  curve  B,  which,  although  a 
load  curve,  has  too  small  an  area  to  generate  the  required  volt- 
age. The  amount  by  which  the  ordinates  of  curve  F  should  be 
increased  may  be  found  by  trial,  but  it  is  generally  possible  to 
estimate  the  necessary  correction  to  give  the  required  result. 


138          PRINCIPLES  OF  ELECTRICAL  DESIGN 

The  added  field  ampere-turns  may  be  thought  of  as  compensat- 
ing for  two  distinct  effects : 

1.  The  loss  of  pressure  due  to  armature  reaction. 

2.  The  loss  of  pressure  due  to  IR  drop  in  armature,  brush 
contacts,  and  series  field  windings. 

The  correction  for  (1)  consists  in  bringing  the  total  flux  up 
to  its  value  on  open  circuit.  It  is  the  ampere-turns  necessary  to 
raise  the  air-gap  density  from  its  average  value  over  curve  B  to 
its  average  value  over  curve  A.  An  approximate  method  of  mak- 
ing this  correction  is  to  find  on  curve  (a)  of  Fig.  49  (i.e.,  the  curve 
corresponding  to  point  a  at  center  of  pole)  the  ampere-turns 
PbPa  required  to  produce  the  difference  of  flux  density  BbBa 
when  OBb  =  average  flux  density  over  curve  B,  and  OBa  = 
average  flux  density  over  curve  A.  The  correction  for  (2) 
consists  in  increasing  the  flux  per  pole  to  such  an  extent  that  it 
will  generate  the  increased  voltage.  This  is  the  correction  for 
compounding;  it  compensates  for  all  internal  loss  of  pressure, 
and  must  include  over-compounding  if  this  is  called  for. 

If  E'  =  required  full-load  developed  volts,  and 

E  =  open-circuit  terminal  volts  (being  also  the   no-load 
developed  volts),  then, 

Area  of  flux  curve  C    _  E' 
Area  of  flux  curve  A  ""  E 

In  this  manner  the  required  area  of  curve  C  is  obtained.  The 
average  flux  density  must  be  increased  in  this  proportion,  and 
the  necessary  additional  ampere-turns  for  this  correction  are 
arrived  at  approximately  by  making  PaPc  (as  indicated  on 
Fig.  49)  such  as  to  increase  the  air-gap  density  in  the  ratio  of 
E'  to  E. 

The  area  of  the  final  curve  C,  which  should  be  measured  for 
the  purpose  of  checking  with  the  required  area,  is  that  com- 
prised between  the  points  on  the  armature  surface  which  corre- 
spond with  the  position  of  the  brushes  on  the  commutator, 
portions  of  the  flux  curve  measured  below  the  datum  line  being 
considered  negative,  and  deducted  from  the  area  measured 
above  the  datum  line. 

The  amount  by  which  the  area  of  the  full-load  flux  curve  C 
must  exceed  the  area  of  the  open-circuit  flux  curve  A  may  be 
determined  approximately  by  estimating  the  probable  voltage 
drop  in  the  series  winding  (if  any)  and  at  the  brush-contact 


FLUX  DISTRIBUTION  139 

surfaces.  In  the  case  of  compound-wound  machines,  it  will 
be  known  at  the  outset  whether  the  dynamo  is  to  be  flat-com- 
pounded or  over-compounded.  Over-compounding  is  resorted 
to  when  the  drop  in  the  circuit  fed  by  the  machine  is  likely  to 
be  high.  The  terminal  voltage  may  then  be  5  per  cent.,  or  even 
10  per  cent.,  higher  at  full  load  than  on  open  circuit.  The  balance 
of  the  e.m.f.  to  be  developed  at  full  load  consists  of: 

1.  The  IR  drop  in  armature  winding. 

2.  The  IR  drop  in  series  field  (if  any). 

3.  The  IR  drop  in  interpole  winding  (if  any). 

4.  The  IR  drop  at  brush-contact  surface. 

Item  (1)  can  readily  be  calculated  since  the  armature  winding 
has  been  designed.  Item  (2)  may  be  estimated  at  from  one- 
fourth  to  one-half  the  armature  drop.  Item  (3)  may  be  esti- 
mated at  from  one-fourth  to  one-half  the  armature  drop.  Item 
(4)  will  be  discussed  later,  but  it  may  be  estimated  at  2  volts, 
and  is  practically  constant  for  machines  of  widely  different 
voltages  and  outputs. 

By  totalling  these  items  of  internal  loss  of  pressure,  and 
adding  thereto  the  required  difference  between  the  terminal 
volts  at  full  load  and  at  no  load,  the  full-load  developed  voltage 
E'  is  obtained,  and  the  required  area  of  curve  C  is  therefore: 

Ef 
Area  of  open-circuit  flux  curve  A  X  -j* 

where  E  is  the  open-circuit  terminal  voltage  as  previously 
defined. 

The  ampere-turns  necessary  to  produce  the  curve  C  of  this 
particular  area  will  not  be  far  short  of  the  total  ampere-turns 
on  the  field  coils  at  full  load,  because  the  air  gap,  teeth  and  slots 
have  considerably  greater  reluctance  than  the  remainder  of  the 
magnetic  circuit.  The  extra  ampere-turns  required  to  over- 
come the  reluctance  of  the  armature  core,  magnet  limbs  and 
frame  will  be  considered  later  when  dealing  with  the  magnetic 
circuit  as  a  whole  and  the  field-magnet  windings. 


CHAPTER  VIII 
COMMUTATION 

44.  Introductory.  —  A  continuous-current  dynamo  is  pro- 
vided with  a  commutator  in  order  that  unidirectional  currents 
may  be  drawn  from  armature  windings  in  which  the  current 
actually  alternates  in  direction  as  the  conductors  pass  successively 
under  poles  of  opposite  kind. 

As  each  coil  in  turn  passes  through  the  zone  of  commutation, 
it  is  short-circuited  by  the  brush,  and  during  the  short  lapse  of. 
time  between  the  closing  and  the  opening  of  this  short-circuit 
the  current  in  the  coil  must  change  from  a  steady  value  of  -f-7c 
to  a  steady  value  of  —  Ic. 

Let  W  =  surface  width  of  brush  (brush  arc)  in  centimeters. 
M  =  thickness  of  insulating  mica  in  centimeters. 
Vc  =  surface  velocity  of  commutator  in  centimeters  per 
second. 

The  time  of  commutation,  in  seconds,  may  then  be  written, 

W  -  M 


Vc 


Since  M  is  usually  small  with  reference  to  W,  it  is  generally 

W 

possible  to  express  the  time  of  commutation  as  tc  =  ^  •  that  is 

r    C 

to  say,  the  time  taken  by  any  point  on  the  commutator  surface 
to  pass  under  the  brush  is  approximately  the  same  as  the  dura- 
tion of  the  short-circuit.  It  is  during  this  time,  tc,  that  the 
current  in  the  commutated  coil  must  pass  through  zero  value 
in  changing  from  the  full  armature  current  of  value  +/c  to  the 
full  armature  current  of  value  —  7C.  If  R  is  the  resistance  of 
the  short-circuited  coil,  and  if  any  possible  disturbing  effect  of 
brush-contact  resistance  be  neglected,  it  is  evident  that  the 
e.m.f.  in  the  coil  should  be  e  —  Ic  X  R  at  the  commencement  of 
commutation.  At  the  instant  of  time  when  the  current  is 
changing  its  direction  —  i.e.,  when  no  current  is  flowing  in  the 
coil  —  the  e.m.f.  is  e  =  0  X  R  =  0.  At  the  end  of  the  time 

140 


COMMUTATION  141 

tc,  when  the  coil  is  just  about  to  be  thrown  in  series  with  the 
other  coils  of  the  armature  winding  carrying  a  current  of  —Ie 
amp.,  the  e.m.f.  in  the  coil  should  be  e  =  —ICR.  It  is  when  the 
e.m.f.  in  the  coil  has  some  value  other  than  this  theoretical  value 
that  sparking  is  liable  to  occur. 

The  theoretical  investigation  of  commutation  phenomena  is 
admittedly  difficult,  because  it  is  almost  impossible  to  take 
account  of  the  many  causes  which  lead  to  sparking  at  the  brushes. 
Some  of  the  problems  to  be  dealt  with  are  of  a  purely  mechanical 
nature,  and  it  is  necessary  to  make  certain  assumptions  and  to 
disregard  certain  influencing  factors  in  order  that  the  essential 
features  of  the  problem  of  commutation  may  be  studied.  The 
writer  has  deliberately  departed  from  the  usual  method  of 
treating  this  subject  because  he  believes  that  it  is  possible  to 
put  the  fundamental  principles  involved  into  a  somewhat  simpler 
form  than  they  are  likely  to  assume  when  clothed  in  mathe- 
matical symbolism.  An  attempt  will  be  made  to  obtain  a 
clear  conception  of  the  physical  phenomena  involved  in  the 
theory  of  commutation. 

Before  the  publication  of  MR.  LAMME'S  paper1  the  methods 
of  DR.  STEiNMETz2  and  DR.  E.  ARNOLD'  formed  the  nucleus  around 
which  the  bulk  of  our  commutation  literature  clung.  MR. 
LAMME'S  paper  has  the  great  merit  of  putting  the  more  or  less 
familiar  problems  of  commutation  in  a  new  light.  The  end  he 
attains  is  approximately,  the  same  as  that  attained  by  any 
other  reasonably  accurate  method  of  analysis,  provided  all 
factors  of  importance  are  included,  and  the  difficulties  he  en- 
counters are  of  the  same  order  and  magnitude  as  those  en- 
countered by  other  investigators;  but,  by  getting  nearer  to  the 
true  physical  conditions  in  the  zone  of  commutation,  he  saves 
us  from  drifting,  sometimes  aimlessly,  on  a  sea  of  abstract  specu- 
lation. Although  the  presentation  of  the  subject  as  given  in 
this  chapter  has  undoubtedly  been  suggested  by  the  reading  of 
MR.  LAMME'S  paper,  yet  its  aim  is  not  so  much  to  furnish  addi- 
tional material  for  the  designer  as  to  give  the  student  a  clear 
conception  of  the  phenomena  of  commutation.  The  writer's 
end  is  simplicity  or  clearness,  even  if  the  less  important  factors 

1  B.  G.  LAMME:  "A  Theory  of   Commutation   and   Its   Application  to 
Interpole  Machines,"  Trans.  A.  I.  E.  E.,  vol.  XXX,  pp.  2359-2404. 

2  "Theoretical  Elements  of  Electrical  Engineering." 
8  "Die  Gleichstrom-Maschine." 


142          PRINCIPLES  OF  ELECTRICAL  DESIGN 

are  entirely  ignored,  while,  in  MR.  LAMME'S  own  words,  his 
method  of  analysis,  including  as  it  does  more  conditions  than 
are  usually  included,  "  instead  of  making  the  problem  appear 
simpler  than  formerly  .  .  .  makes  the  problem  appear  more 
complex."1 

In  the  first  place,  it  may  be  stated  that  considerations  of  a 
mechanical  nature,  such  as  vibration,  uneven  or  oily  commutator 
surface,  insufficient  or  excessive  brush  pressure,  etc.,  cannot  be 
dwelt  upon  here,  and,  in  the  second  place,  ideal  or  "  straight- 
line"  commutation  will  be  assumed,  and  the  conditions  neces- 
sary to  produce  this  —  generally  desirable  —  result  investigated, 
in  order  that  a  multitude  of  more  or  less  arbitrary  assumptions 
may  not  obscure  the  problem  in  its  early  stages.  By  working 
from  the  simplest  possible  case  to  the  more  complex  it  is  thought 
that  the  object  in  view  —  a  physical  conception  of  commutation 
phenomena  leading  to  practical  ends  —  will  best  be  served,  and 
influencing  factors  of  relatively  small  practical  importance  will 
be  either  disregarded  or  but  briefly  referred  to. 

45.  Theory  of  Commutation.  —  Consider  a  closed  coil  of  wire 
of  Tc  turns  moving  in  a  magnetic  field.  At  the  instant  of  time 
t  =  0  the  total  flux  of  induction  passing  through  the  coil  is 
+  $o  maxwells,  and  at  the  instant  of  time  t  =  tc  sec.  the  total 
flux  through  the  coil  is  +  $t  maxwells.  Then  on  the  assumption 
that  the  flux  links  equally  with  every  turn  in  the  coil,  the  average 
value  of  the  e.m.f.  developed  in  the  coil  during  the  interval  of 
time  tc  is 

($f-  $,) 
Em=  - 


If  R  is  the  ohmic  resistance  of  the  coil  and  e  is  any  instantaneous 
value  of  the  e.m.f.  produced  by  the  cutting  of  the  actual  magnetic 
field  in  the  neighborhood  of  the  wire,  the  instantaneous  value 

p 

of  the  current  in  the  coil  is  i  =  g,  because  e  is  the  only  e.m.f. 

in  the  circuit  tending  to  produce  flow  of  current.  The  usual 
conception  of  a  distinct  flux  due  to  the  current  i  producing  a 
certain  flux  linkage  known  as  the  self-inductance  of  the  circuit 
is  avoided;  but  its  equivalent  has  not  been  overlooked,  seeing 
that  the  magnetomotive  force  due  to  the  current  in  the  coil  is 
a  factor  in  the  production  of  the  flux  actually  linked  with  this 
current  at  the  instant  of  time  considered.  It  is  not  suggested 

1  Reply  to  discussion,  Trans.  A.  I.  E.  E.,  vol.  XXX,  p.  2426  (1911). 


COMMUTATION  143 

that  the  orthodox  method  of  introducing  self-induction  and 
mutual  induction  as  separate  entities  endowed  with  certain  prop- 
erties peculiarly  their  own  is  not  without  advantages  in  the 
solution  of  many  problems,  especially  when  mathematical  analysis 
is  resorted  to,  but  it  tends  to  obscure  the  issue  when  seeking  a 
clear  understanding  of  the  physical  aspects  of  commutation. 
The  splitting  up  of  the  magnetic  induction  resulting  from  dif- 
ferent causes  into  several  components  is  frequently  convenient 
and  should  not  be  condemned  except  in  certain  cases  when  iron 
is  present  in  the  magnetic  circuit.  It  cannot,  however,  be  de- 
nied that  self-induction  and  mutual  induction  are  frequently 
thought  of  as  different  from  other  kinds  of  induction.  We  are 
indebted  for  this  state  of  things  to  some  writers  whose  familiarity 
with  mathematical  methods  renders  a  clear  physical  conception 
of  complicated  phenomena  unnecessary,  but  the  practical  engi- 
neer or  designer  who  produces  the  best  work,  especially  in  de- 
partures from  standard  practice,  is  usually  he  who  has  the  clearest 
vision  of  the  physical  facts  involved  in  the  problem  under  con- 
sideration. If  the  term  self-induction  calls  up  a  mental  picture 
of  magnetic  lines,  being  a  certain  component — expressed  in 
maxwells — of  the  total  or  resultant  flux  of  induction  in  a  circuit, 
this  does  not  prevent  our  speaking  of  flux  linkage  per  ampere 
of  current  as  inductance — expressed  in  henrys — and  using  the 

formula  e  =  L  -j.  to  calculate  that  component  of  the  total  e.m.f. 

in  a  circuit  which  would  have  a  real  existence  if  the  field  due  to 
the  current  i  in  the  wire  were  alone  to  be  considered. 

Following  the  lead  of  MR.  LAMME,  the  wires  in  the  coil  under- 
going commutation  will  be  thought  of  as  cutting  through  a  total 
flux  of  induction,  expressed  in  magnetic  lines  or  maxwells,  this 
flux  being  the  result  of  the  magnetizing  forces  of  field  coils  and 
armature  windings  combined. 

In  Fig.  54  the  thick-line  rectangle  represents  a  full-pitch 
armature  coil  of  T  turns  undergoing  commutation.  The  dotted 
rectangles  show  the  position  of  two  consecutive  field  poles,  and 
the  shaded  curve  represents  the  ascertained  or  calculated  flux 
distribution  over  the  armature  surface.  The  ordinates  of  this 
curve  indicate  at  any  point  on  the  periphery  the  density  of  the 
flux  entering  the  armature  core.  The  direction  of  slope  of 
the  shading  lines  indicates  whether  the  flux  is  positive  or 
negative.  A  method  that  may  be  followed  in  predetermining 


144 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


the  flux  distribution  over  the  armature  surface,  including  the 
interpolar  space,  was  outlined  in  Chap.  VII  (refer  to  Art.  41, 
42  and  43) ,  and  the  flux-distribution  curves  of  Fig.  54  might  have 
been  obtained  in  the  same  manner  as  the  flux  curve  C  referred  to 
in  Art.  43.  The  coil  of  Fig.  54  is  supposed  to  be  moving  from 
left  to  right,  and  measurements  on  the  horizontal  axis  XX  may 
represent  either  distance  travelled  or  lapse  of  time,  since  the  arma- 
ture is  revolving  at  a  uniform  speed.  The  case  considered  is 
that  of  a '  dynamo  without  commutating  poles,  with  brushes 
moved  forward  from  the  geometric  neutral  or  no-load  com- 
mutation position  until  a  neutral  commutating  zone  is  again 


FIG.  54. — Diagram   showing   ideal  armature  coil  in   commutating   zone. 

found.  The  flux  curves  as  drawn  are  the  result  of  the  combined 
m.m.fs.  of  field  coils  and  armature  windings.  During  the  time 
of  commutation,  tc,  which,  if  we  neglect  the  effect  of  mica  thick- 
ness, is  the  time  taken  by  a  point  on  the  commutator  to  pass 
under  the  brush  of  width  W,  the  conductors  on  the  right-hand 
side  of  the  short-circuited  coil  have  been  moved  through  the 
neutral  zone  from  a  weak  field  of  positive  polarity  into  a  weak 
field  of  negative  polarity,  while  the  conductors  on  the  left-hand 
side  of  the  coil  have  moved  from  a  weak  field  of  negative  polarity 
into  a  weak  field  of  positive  polarity.  Owing  to  the  symmetry 
of  the  fields  under  the  poles  of  opposite  kind  (i.e.,  the  similarity 
in  shape  and  equality  in  magnitude  of  the  shaded  flux  curves), 
and  the  fact  that  the  small  portions  of  the  flux  curves  near  the 


COMMUTATION  145 

neutral  point  may  be  considered  as  straight  lines,  the  resultant 
flux  cut  by  the  two  coil-sides — joined  in  series  by  the  end  con- 
nections— may  be  represented  by  the  shaded  area  in  Fig.  55, 
where  positive  values  are  measured  above,  and  negative  values 
below,  the  horizontal  axis.  Intervals  of  time  are  measured 
horizontally  from  left  to  right,  and  the  straight  line  BE'  rep- 
resents the  flux  distribution  in  the  commutating  zone.  The 
direction  of  this  flux  is  such  as  to  develop  in  the  short-circuited 
coil,  at  every  instant  of  time  during  the  period  of  commutation, 
an  e.m.f.  tending  to  produce  a  current  in  the  required  direction; 
that  is  to  say,  from  the  commencement  of  short-circuit,  when 

+£• 

*^J  >^t 

t-te 


! 

FIG.  55. — Flux  distribution  in  commutating  zone  of  ideal  armature  coil. 

t  =  0,  until  the  middle  of  the  commutation  period,  when  both 
flux  and  current  are  of  zero  value,  the  small  amount  of  flux  cut 
by  the  short-circuited  conductors  is  of  the  same  kind  as  that 
previously  cut  by  the  conductors,  while  from  the  time  t  =  tc/2 
until  the  end  of  commutation  (t  —  tr)  the  flux  is  of  the  opposite 
kind,  being  such  as  will  cause  the  current  to  flow  in  the  oppo- 
site direction.  The  amount  of  the  flux  required  to  bring  about 
this  condition  is  only  a  small  percentage1  of  the  flux  cut  by  a  coil 
under  the  main  poles  in  the  same  interval  of  time,  because  the 
resistance  of  the  armature  windings  is  always  low  in  comparison 
with  the  resistance  of  the  external  circuit,  and,  as  a  matter  of 
fact,  it  is  the  average  value  of  the  flux  entering  the  armature 
over  the  commutating  zone  with  which  the  designer  is  usually 
concerned.  If  the  brushes  are  so  placed  as  to  bring  the  short- 
circuited  conductors  in  a  neutral  field,  satisfactory  commutation 
will  result. 

1  The  flux  density  where  the  coil-side  enters  or  leaves  the  commutating 
zone  (the  positions  t  =  0  and  t  =  k  of  Fig.  55)  would  be  about  2.5  per  cent, 
of  the  average  density  under  the  main  poles,  because  this  is  the  ratio  of  the 
armature  IR  drop  to  the  developed  voltage  in  a  well-designed  dynamo  of 
moderate  size — say,  50  to  100  kw. 
10 


146 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


Returning  to  a  consideration  of  the  case  represented  by  Fig. 
54,  it  must  not  be  overlooked  that  the  armature  coil  there  shown 
is  not  of  a  practical  shape,  the  end  connections  are  shown  parallel 
to  the  direction  of  travel  of  the  coil,  and  the  cutting  of  fluxes  by 
these  end  portions  of  the  coil  has  not  been  considered.  When 
we  consider  the  end  fluxes,  or  the  effect  of  commutating  inter- 
poles,  especially  when  these  are  not  equal  in  number  to  the  main 
poles  or  do  not  extend  the  full  length  of  the  armature  core,  then 
the  flux  cut  by  the  short-circuited  conductors  at  any  given  part 
of  their  total  length — such  as  the  center  of  the  " active"  portion, 
whether  on  a  smooth  core  or  in  slots — may  have  an  appreci- 
able value;  but  if  we  consider  the  total  flux  cut  by  all  parts  of 
the  wire  forming  the  commutated  coil,  when  the  current  i  in  this 
coil  is  passing  through  zero  value,  it  is  most  emphatically  true 
that  the  coil  as  a  whole  is  moving  in  a  "  neutral  field,"  i.e.,  a 
resultant  field  which  is  either  of  zero  value  (when  the  sum  of  all 
its  components  is  correctly  taken)  or  of  which  the  direction  is 
parallel  to  the  direction  of  travel  of  the  conductors. 


FIG.  56. — Diagram  of  coil  and  commutator  during  commutation. 


At  the  beginning  and  end  of  the  commutation  period  the 
field  in  which  the  coil  moves  should  be  such  as  to  produce  an 
e.m.f.  in  the  short-circuited  coil  of  the  value  e  =  ICR,  where  Ic 
is  the  value  of  the  current  per  path  of  the  armature  circuit  and 
R  is  the  resistance  of  the  short-circuited  coil.  On  the  assumption 
of  a  uniform  current  density  over  the  surface  of  the  brush,  the 
brush  contact  resistance  need  not  be  taken  into  account,  as 
will  be  clear  from  the  following  considerations.  Fig.  56  shows  a 
brush  of  width  W  covering  several  segments  of  the  commutator. 
The  total  current  entering  the  brush  is  2/c,  and  since  the  density 


COMMUTATION  147 

is  constant  over  the  surface  of  contact,  the  current  entering  the 

q 

brush  through  any  surface  of  width  S  is  2/c  X  yy'     To  cal- 

culate the  volts  e  that  must  be  developed  in  the  coil  of  resistance 
R  when  the  distance  yet  to  be  travelled  before  the  end  of  com- 
mutation is  w,  consider  the  sum  of  the  potential  differences  in 
the  local  circuit  AabB  which  is  closed  through  the  material  of 
the  brush.  This  leads  to  the  equation 

e  =  iR  +  ibRb  -  iaRa  (67) 

where  Ra  and  Rb  are  contact  resistances  depending  upon  the 
areas  of  the  surfaces  through  which  the  current  enters  the  brush. 
Under  the  conditions  shown  in  Fig.  56,  the  contact  surfaces  Sa 
and  Sb  are  equal,  and  the  currents  ia  and  4  are  therefore  also 
equal.  It  follows  that  the  voltage  drops  iaRa  and  ibRb  are  equal 
and  cancel  out  from  equation  (67).  The  same  is  true  in  the 
later  stages  of  commutation  when  Sa  is  no  longer  equal  to  Sb 
but  to  the  portion  w  of  the  brush  which  remains  in  contact  with 
the  segment  A.  The  relations  between  the  currents  and  the 
surface  resistances  are  then  obtained  by  expressing  these  quan- 
tities in  terms  of  the  contact  surface,  thus: 

ia  =  w  X  k\ 

ii  =  Sb  X  &i 

R.  =  ~  X  k> 

Rb    =   ~cT    X   &2 
06 

where  k\  and  k2  are  constants,  and  the  voltage  drop  iaRa  is  seen 
to  be  still  equal  to  the  drop  ibRb.  It  follows  that  the  only  e.m.f. 
to  be  developed  in  the  short-circuited  coil  when  uniform  current 
distribution  is  required  will  be  e  =  iR. 

The  instantaneous  value  (i)  of  the  current  in  the  coil  under- 
going commutation  can  be  expressed  in  terms  of  the  brush  width 
W  and  the  distance  (w)  through  which  the  coil  still  has  to  travel 
before  completion  of  commutation,  because, 

i  =  Ic  -  2IC  X 
lc 


(68) 


W 
and 


148          PRINCIPLES  OF  ELECTRICAL  DESIGN 

At  the  beginning  and  end  of  commutation,  when  w  is  equal  to 
W  or  to  zero,  the  maximum  value  of  the  required  voltage  is 

6    =   ICR. 

In  this  study  of  the  voltage  to  be  developed  in  the  coil  under- 
going commutation  in  order  to  produce  a  uniform  current 
distribution  over  the  brush  surface,  the  resistance  of  the  brush 
itself  has  been  considered  negligible;  but  with  the  assumption 
of  a  uniform  current  distribution  over  the  cross-section  of  the 
brush  the  actual  resistance  of  the  brush  material,  even  if  it 
is  relatively  high,  will  not  appear  as  a  modifying  factor  in  the 
general  formula  (68). 

Referring  again  to  Fig.  55,  if  the  flux  curve  EB'  may  be  con- 
sidered a  straight  line,  the  current  \i  =  ^}  will  also  obey  a 
straight-line  law.  It  will  fall  from  the  value  +7C  to  zero  in 
the  time  ^  and  rise  again  to  the  value  —  Ic  at  the  end  of  the 

period  tc,  according  to  the  simple  law  expressed  by  the  straight 
line  in  Fig.  55.  If  the  change  of  current  actually  occurs  in  this 
manner,  we  have  what  is  called  "  straight-line "  or  ideal  com- 
mutation. The  commutation  is  then  ideal  or  perfect,  not  only 
because  it  relieves  the  designer  of  much  intricate  and  discouraging 
mathematical  work,  but  because  it  is  the  only  means  by  which 
the  current  density  can  be  maintained  constant  over  the  brush 
surface  of  the  usual  rectangular  form.  It  is  generally  the  aim  of 
the  designer  to  maintain  this  current  density  as  nearly  constant 
as  possible,  because  unequal  current  density  leads  to  local  varia- 
tions of  temperature  and  resistance  in  the  carbon  brush,  and  in 
those  parts  where  the  density  attains  very  high  values  the  ex- 
cessive heating  leads  to  pitting  of  the  commutator  surface  even 
if  visible  sparking  does  not  occur.  Whatever  method  of  study- 
ing commutation  phenomena  is  followed,  it  is  usual  to  assume 
some  law  connecting  the  variable  current  i  with  the  time  t 
and  then  investigate  the  causes  which  will  bring  about  this 
condition.  The  straight-line  law  will  therefore  be  assumed,  but 
the  thing  of  immediate  moment — being  in  fact  the  whole  problem 
of  commutation  in  its  broader  aspect — is  the  location,  or  the 
creation,  of  a  neutral  zone  where  the  actual  resultant  flux  cut  by 
the  coil  undergoing  commutation  will  be  zero. 

Although  the  assumption  of  a  smooth-core  armature  very 
greatly  simplifies  the  problem,  especially  when  an  effort  is  made 


COMMUTATION  149 

to  picture  the  actual  distribution  of  the  magnetic  flux,  it  seems 
preferable  to  consider  a  machine  with  toothed  armature  because 
this  is  the  case  which  has  generally  to  be  dealt  with  by  the 
practical  designer,  and  moreover  it  is  exactly  this  question  of 
teeth,  or  what  is  known  as  the  slot  flux,  which  sometimes  leads 
to  confusion  of  ideas,  if  not  to  inaccurate  conclusions,  and  it 
should  therefore  not  be  disregarded  in  any  modern  theory  of 
commutation. 

46.  Effect  of  Slot  Flux. — In  Fig.  57  an  attempt  has  been  made 
to  represent,  by  the  usual  convention  of  magnetic  lines,  the  flux 
due  to  the  armature  current  alone,  which  enters  or  leaves  the 
armature  periphery  in  the  interpolar  space  when  the  field  mag- 
nets are  unexcited.  The  position  chosen  for  the  brushes  is  the 
geometric  neutral — i.e.,  the  point  midway  between  two  (sym- 
metrical) poles — and  the  magnetic  lines  leaving  the  teeth  will 
cross  the  air  spaces  between  armature  surface  and  field  poles 
and  so  close  the  magnetic  circuit.  The  brush  is  supposed  to 
cover  an  angle  equal  to  twice  the  slot  pitch:  the  current  in  the 
conductor  just  entering  the  left-hand  end  of  the  brush  is  -f-/c, 
the  current  in  the  conductor  under  the  center  of  brush  is  zero, 
and  the  current  in  the  conductor  just  leaving  the  brush  on  the 
right-hand  side  is  —  Ic.  The  armature  is  supposed  to  be  rotat- 
ing, and  it  will  be  seen  that  the  conductors  in  which  the  current 
is  being  commutated  are  cutting  the  flux  set  up  by  the  armature 
as  a  whole.  It  is  important  to  note  that  the  flux  cut  by  a  con- 
ductor while  travelling  between  the  two  extreme  positions  during 
which  the  short-circuit  obtains  is  not  only  the  flux  passing  into 
the  air  gaps  from  the  tops  of  the  teeth  included  between  these 
extreme  positions  of  the  conductor,  but  includes  also  the  flux 
due  to  the  currents  in  the  short-circuited  conductors,  which 
crosses  the  slot  above  the  conductor1  and  leaves  the  armature 
surface  by  teeth  which  are  not  included  in  what  at  first  sight  may 
seem  to  be  the  commutating  zone.  In  other  words,  the  portion 
of  the  armature  flux  cut  by  a  conductor  undergoing  commuta- 
tion when  no  reversing  flux  is  provided  from  outside  is  that 
which  passes  up  through  the  roots  of  the  teeth  included  between 
the  two  extreme  positions  of  the  short-circuited  coil.  This 
picture  of  the  conductor  cutting  the  field  set  up  by  the  armature 

1  For  the  sake  of  simplicity,  a  single  conductor  is  shown  at  the  bottom  of 
each  slot  and  the  whole  of  the  slot  flux  is  supposed  to  link  with  it.  The 
calculation  of  the  "equivalent"  slot  flux  will  be  taken  up  later. 


150 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


currents  is  especially  useful  when  calculations  are  made,  as  will 
frequently  be  found  convenient,  by  considering  the  separate 


FIG.  57. — Flux  in  commutating  zone  due  to  armature  m.m.f.  only. 

component  fluxes  due  to  distinct  causes,  all  combining  to  pro- 
duce the  actual  or  resultant  flux.  It  is  not  difficult  to  see  that 
the  flux  shown  in  Fig.  57  is  never  such  as  to  generate  an  e.m.f. 


FIG.  58. — Flux  in  commutating  zone  near  leading  pole  tip. 

tending  to  reverse  the  current  in  the  short-circuited  conductor. 

Consider  now  Fig.  58.     The  main  poles  have  been  excited 

and  the  brushes  moved  forward  until  a  satisfactory  commutating 


COMMUTATION  151 

zone  has  been  found  where  the  fringe  of  flux  from  the  leading 
pole  tip  is  sufficient  to  neutralize  the  flux  due  to  the  armature 
windings. l  With  the  excitation  of  the  leading  main-pole  tending 
to  send  flux  through  the  armature  core  from  right  to  left,  and  the 
armature  e.m.f.  tending  to  produce  a  flux  distribution  gener- 
ally as  indicated  in  Fig.  57,  the  resulting  flux  distribution  in  the 
commutating  zone  will  be  somewhat  as  shown  in  Fig.  58.  Here 
the  flux  cut  by  the  conductors  during  cpmmutation  is  represented 
by  eight  lines  only,  the  direction  of  this  commutating  flux 
being  such  as  to  maintain  the  current  during  the  earlier  stages 
of  commutation  and  reverse  it  during  the  later  stages.  At  a 
point  midway  between  the  two  extreme  positions  the  conductor 
is  cutting  no  flux,  and  the  current  is  therefore  zero.  It  should  be 
observed  that  the  correct  position  for  the  brush  is  in  advance  of 
the  "apparent"  neutral  zone;  that  is  to  say,  the  position  of  the 
neutral  field  on  the  surface  of  the  armature  does  not  correspond 
with  the  correct  position  for  the  center  of  the  brush.  That  is 
because  the  slot  flux  must  enter  through  the  upper  part  of  the 
teeth  if  it  is  not  to  be  cut  by  the  conductors  during  commutation. 

Thus  the  conductor,  which  at  the  instant  of  time  t  =  ^  must  be 

a 

in  a  neutral  field,  is  actually  below  a  point  on  the  armature 
periphery  where  flux  is  entering  or  leaving  the  teeth,  and  this 
condition  occurs  even  when,  as  in  the  present  instance,  the  effect 
-of  the  end  connections  is  entirely  negligible.  This  flux,  which 
enters  the  teeth  comprised  between  the  two  extreme  positions 
of  the  short-circuited  conductors,  is  neither  more  nor  less  than 
the  slot  flux  (or  equivalent  slot  flux,  as  the  case  may  be).  It  is 
represented  by  12  lines  in  the  diagram  Fig.  58,  and  it  must 
be  provided  by  the  leading  pole  shoe  if  brush  shift  is  resorted 
to,  or  by  the  commutating  interpole  when  this  method  of 
cancelling  the  armature  flux  is  adopted. 

47.  Effect  of  End  Flux.— When  the  effect  of  the  end  connec- 
tions of  the  armature  coils  cannot  be  neglected,  the  armature 
flux  cut  by  this  portion  of  the  short-circuited  coil  must  be 
cancelled  in  the  same  manner  as  the  slot  flux;  that  is  to  say,  an 

1  Credit  is  given  the  reader  for  the  ability  to  read  in  the  expression  "a 
flux  neutralizing  a  flux"  the  more  scientific  but  less  convenient  expression 
"the  magnetic  force  due  to  one  magnetizing  source  being  of  such  magnitude 
and  direction  as  to  neutralize  the  magnetic  force  due  to  a  second  magnetizing 
source,  thus  causing  the  resultant  flux  of  induction  to  be  of  zero  value." 


152 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


equal  amount  of  flux,  but  of  opposite  sign,  must  enter  the  arma- 
ture from  the  leading  pole  tip  or  interpole,  and  this  component 
of  the  compensating  flux  will  actually  be  cut  by  the  conductors 
in  the  slot,  thus  neutralizing  the  e.m.f.  developed  by  the  cutting 
of  the  end  fluxes.  This  will  be  made  clearer  by  reference  to  Fig. 
59,  which  is  generally  similar  to  Fig.  54,  except  that,  instead  of 


FIG.  59. — Diagram  showing  component  fluxes  cut  by  coil  during 
commutation. 

the  actual  interpolar  flux,  two  distinct  curves  have  been  drawn, 
the  one,  F,  representing  flux  distribution  over  armature  peri- 
phery due  to  the  field  coils  acting  alone,  and  the  other,  Z,  repre- 
senting the  flux  distribution  due  to  the  armature  windings  acting 
alone.  The  addition  of  these  two  fluxes  at  every  point  will  not 
always  reproduce  the  actual  flux  curve  of  Fig.  54,  because  of 
possible  saturation  of  portions  of  the  iron  circuit  such  as  the 
armature  teeth  and  pole  tips;  but,  in  the  commutating  zone,  the 
method  of  adding  the  several  imaginary  components  of  the 


COMMUTATION  153 

actual  flux  is  not  objectionable,  and  the  active  conductors  AA' 
in  Fig.  59  may  be  considered  as  moving  in  a  field  of  which  the 
density  is  represented  by  the  length  MN,  since  the  portion  of 
the  field  flux  represented  by  the  distance  between  the  point 
N  and  the  datum  line  is  neutralized  by  the  armature  flux  at 
this  point.  Let  ABC  represent  the  position  of  the  end  connec- 
tions of  the  coil  undergoing  commutation,  then  the  portion  AB  is 
cutting  end  flux  due  to  the  armature  currents  in  all  the  end  con? 
nections,  and  the  direction  of  this  flux  will  be  the  same  as  that 
represented  by  the  curve  Z,  all  as  indicated  by  the  direction  of  the 
shading  lines.  The  portion  BC  of  the  short-circuited  coil  will 
be  cutting  flux  of  the  same  nature  as  the  armature  flux  cut  by  the 
slot  conductors  CC',  and  the  e.m.f.  due  to  the  cutting  of  the  end 
fluxes  will  be  of  the  same  sign  as  that  due  to  the  cutting  of  the 
armature  flux  Z;  that  is  to  say,  it  will  tend  to  oppose  the  reversal 
of  current  and  must  therefore  be  compensated  for  by  a  greater 
brush  lead  or  a  stronger  commutating  pole.  Similar  arguments 
apply  to  the  end  connections  A'B'C'  at  the  other  end  of  the 
armature.  A  means  of  calculating  the  probable  value  of  the 
effective  end  flux  will  be  considered  later;  but  for  the  present 
it  may  be  assumed  that  the  average  value  of  the  density  Be 
of  the  field  cut  by  the  end  connections  is  known.  It  may,  there- 
fore, be  used  for  correcting  the  ordinate  of  the  curve  Z  at  the 
point  0.  Thus,  the  flux  cut  by  the  portion  ABC  of  the  end 
connections  (see  Fig.  59)  in  the  time  tc  is 

$,  =  Be  X  x  X  length  of  ABC 
or 

$>*  =  BeWa  sm  a  X  length  of  ABC 

where  a  is  the  angle  between  the  lay  of  the  end  connections  and 
the  direction  of  travel,  and  Wa  is  the  arc  covered  by  the  brush, 
expressed  in  centimeters  of  armature  periphery.  The  equivalent 
flux  density  Ba  which  has  to  be  cut  by  the  slot  conductors  A  A' 
to  develop  the  same  average  voltage  is  obtained  from  the  relation 
BaWa  X  length  AA'  =  BaWa  sin  a  X  length  ABC 
which  gives 

i,,,,,,-i  i,  A  Jin 

(69) 


or,  if  preferred, 

>  length  AA' 
.   2(BH) 

154          PRINCIPLES  OF  ELECTRICAL  DESIGN 

This  may  be  plotted  in  Fig.  59  as  the  ordinate  OE,  making  NE 
represent  the  armature  flux,  on  the  assumption  that  the  whole 
of  this  flux  component  is  cut  by  the  "active"  portion  of  the 
coil;  and  this  suggests  a  graphical  method  of  locating  the  correct 
brush  position  when  commutating  poles  are  not  used,  because 
what  may  be  called  the  equivalent  neutral  zone  is  found  when 
the  conductor  A  A'  occupies  a  position  such  that  the  length 
NE  is  exactly  equal  to  OM.  If  this  position  cannot  be  found 
without  passing  under  the  tip  of  the  pole  shoe  (represented  by 
the  heavy  dotted  rectangle),  the  machine  will  not  commutate 
perfectly  without  the  addition  of  a  commutating  interpole. 

The  question  of  relative  magnitude  of  these  end  flux  e.m.fs. 
deserves  some  attention,  because  it  would  be  foolish  to  compli- 
cate the  problem  of  commutation  if  the  correction,  when  made, 
is  of  little  practical  moment.  It  is  claimed  by  some  writers 
that  refinement  of  analysis  and  calculation  is  always  commend- 
able even  when  built  upon  a  foundation  that  is  admittedly  a 
mere  approximation.  With  this  attitude  of  mind  the  present 
writer  has  no  sympathy;  it  appears  to  lack  the  sense  of  propor- 
tion. Apart  from  any  considerations  of  a  mechanical  nature, 
the  practical  problem  of  commutation,  from  whatever  point  of 
view  it  is  approached,  is,  and  always  will  be,  the  correct  determi- 
nation of  the  field  in  which  the  short-circuited  coil  is  moving, 
whether  this  conception  of  the  magnetic  condition  is  buried 
in  the  symbols  L  and  M ,  and  referred  to  as  inductance,  expressed 
in  henrys,  or  considered  merely  as  any  other  magnetic  field; 
and  it  would  surely  be  a  waste  of  time  and  mental  effort  to  intro- 
duce refinements  if  the  percentage  correction,  when  made,  is  of 
a  small  order  of  magnitude.  The  question  of  end  fluxes,  how- 
ever, is  one  of  real  practical  importance;  the  end  flux  in  actual 
machines  is  not  of  negligible  amount,  and  although  it  cannot 
be  calculated  exactly,  it  is  a  factor  which  should  not  be  left 
out  of  consideration .  It  is  true  that  we  do  not  concern  ourselves 
with  the  end  fluxes  when  calculating  the  useful  voltage  developed 
in  the  active  coils;  but,  apart  from  the  fact  that  in  this  connec- 
tion the  amount  of  the  end  flux  is  relatively  small,  it  is  not  difficult 
to  see  that  the  e.m.f.s  generated  in  the  end  connections  as  they 
cut  through  the  end  fluxes  due  to  the  armature  currents  balance 
or  counteract  each  other  and  have  no  effect  on  the  terminal 
voltage.  The  conception  of  the  end  connections  cutting  through 
the  flux  due  to  the  armature  as  a  whole,  as  indicated  in  Fig.  59, 


COMMUTATION  155 

seems  more  natural,  and  is  more  helpful  to  the  understanding  of 
commutation  phenomena,  than  what  might  be  termed  the 
academic  method,  in  which  more  or  less  reasonable  assumptions 
are  made  in  respect  to  self-  and  mutual  inductances;  but  it  is 
not  suggested  that  the  one  method  is  necessarily  superior  to  the 
other  so  far  as  practical  results  are  concerned.1  While  moving 
from  the  position  at  the  commencement  of  the  commutation 
where  the  current  is  -f-/c  to  the  position  at  the  end  of  commuta- 
tion where  the  current  is  —  It,  the  short-circuited  coil  has  cut 
through  the  flux  of  self-  and  mutual  induction — through  the 
whole  of  it,  not  merely  through  certain  components  of  the  total 
flux  in  the  particular  region  considered.  This  is  well  expressed 
by  MR.  MENGES  when  he  says2:  ".  .  .  Self-induction  is  in 
no  way  distinguishable  from  other  coexistent  electromagnetic 
induction.  Therefore,  when  the  real  magnetic  flux  resulting 
from  all  causes,  and  its  changes  relative  to  a  given  circuit,  are 
taken  into  account,  the  self-induction  is  already  included,  and 
it  would  be  erroneous  to  add  an  e.m.f.  of  self-induction." 

48.  Calculation  of  End  Flux. — With  a  view  to  calculating 
within  a  reasonable  degree  of  accuracy  the  flux  density  in  the 
zone  ABC  of  Fig.  59,  it  is  necessary  to  make  certain  assumptions 
and  to  use  judgment  in  applying  the  calculated  results,  because 
it  is  not  possible  to  determine  this  value  with  scientific  accuracy 
even  when  the  exact  shape  of  the  armature  coils  and  the  con- 
figuration of  the  surrounding  masses  of  iron  are  known. 

In  the  first  place,  the  angle  a  of  Fig.  59,  wil)  be  taken  as  45 
degrees,  which,  although  perhaps  slightly  greater  than  the 
average  on  modern  multipolar  machines,  has  the  advantage 
that  it  permits  us  to  treat  the  wires  AB  and  BC  as  being  at  right 
angles  to  each  other.  The  further  assumption  will  be  made 
that  the  armature  is  of  large  diameter,  and  the  developed  view 
of  the  end  connections,  as  shown  in  Fig.  60,  can  therefore  be 
considered  as  lying  in  the  plane  of  the  paper.  The  flux  in  the 
zone  occupied  by  the  portion  A  B  of  the  coil  undergoing  commu- 

1  As  a  matter  of  fact,  a  careful  study  of  the  problem  will  show  that  the 
total  armature  flux  cut  by  one  commutating  coil  during  the  period  of  short- 
circuit — being  the  difference  between  the  number  of  lines  threading  the  coil 
immediately    before     and    immediately     after    commutation — is    almost 
entirely  due  to  the  changes  of  current  that  have  taken  place  in  the  coils 
under  the  brushes  during  the  period  of  commutation. 

2  C.  L.  R.  E.  MENGES  in  the  London  Electrician,  Feb.  28,  1913. 


156 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


tation  is  due  to  the  currents  in  all  the  neighboring  parallel  con- 
ductors comprised  in  the  parallelogram  A  DEC.  The  direction 
of  flow  of  current  in  these  parallel  conductors  is  indicated  by  the 
arrows,  being  outward  (i.e.,  from  A  to  B)  on  the  left-hand  side 
of  the  commutating  zone,  and  inward  (from  B  to  A)  on  the  right- 
hand  side.  The  intensity  of  the  field  produced  on  AB  by  any 
one  of  these  wires,  if  the  lines  of  force  are  assumed  to  be  circles 
in  a  path  consisting  entirely  of  air  (the  proximity  of  masses  of 
iron  being  for  the  present  ignored),  will  be  inversely  proportional 


FIG.  60. — Developed  view  of  armature  end  connections. 

to  the  distance  of  the  wire  considered;  and  the  extent  to  which 
this  wire  will  be  effective  in  producing  a  field  along  AB  will 
depend  upon  its  length.  Thus,  a  conductor  such  as  EF  will  pro- 
duce not  only  a  stronger  component  of  the  resulting  field  in  the 
commutating  zone  than  the  wire  GH,  but  a  field  of  which  the 
extent  is  measured  by  the  length  EF,  while  the  more  distant 
wire  will  produce  a  weaker  field  over  a  length  equal  to  GH  only. 
Thus  the  effect  of  the  more  distant  wires  in  building  up  the  flux 
over  the  commutating  zone  decreases  very  rapidly  with  increase 
of  distance.  Fig.  61  represents  a  section  perpendicular  to  the 
conductor  AB  of  Fig.  60.  It  is  assumed  that  the  brush  covers 


COMMUTATION 


157 


two  bars  (a  reasonable,  but  not  a  necessary,  assumption),1 
and  the  condition  shown  in  Fig.  61,  corresponds  to  the  middle 
of  the  commutation  period,  with  zero  current  in  the  short-cir- 
cuited coil  and  full  armature  currents  of  +/c  and  —  Ic  respectively 
in  the  neighboring  conductors. 


.-Connection  EF 


<8>  8 


©  © 


\M    !     I 


FIG.  61. — Mangetic  flux  due  to  end  connections. 


Considering  the  full-pitch  coil  of  pitch 
over  the  armature  surface,  we  can  write 

T  =  nX 


centimeters  measured 


where  n  =  number  of  slots  per  pole,  and  X  =  slot  pitch  in 
centimeters.  Then  the  length  AB  (Fig.  60),  which  is  approxi- 
mately one-quarter  of  the  total  length  of  " inactive"  copper 
per  coil,  is 

T          nX 

= 


and  the  pitch  of  the  end  windings  (which  are  supposed  to  lie 
in  the  same  plane  as  the  conductors  in  the  slots)  will  be, 

X          I 
a  = 


Let  T  —  number  of  conductors  or  turns  per  coil  (which  is  not 
necessarily  the  same  as  the  number  of  turns  between  com- 
mutator bars,  because  there  may  be  more  commutator  bars  than 
there  are  slots  on  the  armature),  and  let  Ic  =  the  amperes  of 

1  Within  practical  limits,  the  width  of  the  brush  does  not  appreciably 
affect  the  average  density  of  the  armature  flux  cut  by  the  commutated  coil. 
A  wide  brush,  by  short-circuiting  several  coils,  reduces  the  number  of  con- 
ductors carrying  the  full  armature  current,  and  to  this  small  extent  the  total 
m.m.f.  producing  the  flux  in  the  commutating  zone  is  less  with  a  wide 
brush  than  with  a  narrow  brush. 


158          PRINCIPLES  OF  ELECTRICAL  DESIGN 

current  per  conductor;  then,  since  the  field  intensity  at  a  distance 
y  cm.  from  a  straight  conductor  is 

„       0.2  X  current  in  amperes 
ti  =  — 

y 

we  may  write,  for  field  intensity  due  to  the  group  of  conductors 
EF, 

„         0.2  X  TIC 

H" '-     ~W 

The  flux  produced  in  the  zone  AB  by  the  same  group  of  wires 
(EF)  will  be  proportional  to  the  value  of  H  multiplied  by  the 
length  EF.  Thus,  in  a  zone  1  cm.  wide,  of  which  the  center  line 
is  AB,  the  flux  of  induction  due  to  the  conductor  EF  is 

=  0.277Q      _ 

The  sum  of  all  such  elements  of  the  total  flux,  taken  for  all  the 
parallel  conductors  on  both  sides  of  AB  will  be 

QATIC  [,  I  -  2a       I  -  3a  I  -  i 


which,  bearing  in  mind  the  relation  I  =  an,  simplifies  into 

(70) 


The  total  flux  (maxwells)  cut  by  the  end  connections  ABC, 
being  one-half  of  the  length  of  "  inactive"  copper  in  the  corn- 
mutated  coil,  is  given  by  the  expression 

$e  =  2$  X  Wa  sin  a 

=    VZ$   X    Wa 

=  OAV2TIcnWa   g  +  J  +  1  •      +  J)  (71) 

in  which  Wa  is,  as  before,  the  brush  arc  expressed  in  centimeters 
of  armature  periphery. 

The  value  of  the  series  in  the  brackets  is  readily  computed 
with  the  aid  of  a  table  of  reciprocals,  but  if  preferred  this  series 
can  be  put  in  the  form  (log.  2n)  —  1,  which  is  really  more  ac- 
curate, since  it  assumes  a  current  uniformly  distributed  through 
the  copper  section  of  the  conductors  instead  of  being  concen- 
trated at  the  center  of  each  coil  as  assumed  in  deriving  formula 
(70). 


COMMUTATION  159 

The  assumption  has  been  made  that  the  paths  of  the  flux  lines 
are  air  paths  only;  but  on  account  of  the  proximity  of  the  pole 
shoes  to  the  points  where  the  end  connections  leave  the  slots, 
and  also  because  the  conductors  actually  remain  parallel  to  the 
shaft  for  a  short  distance  beyond  the  core,  the  value  of  3>e  would 
be  larger  than  as  given  by  formula  (71).  A  constant  should 
therefore  be  included,  and  if  the  convergent  series  is  replaced  by 
the  logarithmic  function,  the  formula  becomes, 

$e  =  OA<^2kTIenWa[(lo&  2n)  -  l]  (72) 

The  average  flux  density  over  the  zone  considered  is 

nj-.1]  (73) 


If  the  end  coils  lie  on  a  cylindrical  support  of  iron  or  steel,  the 
reluctance  of  the  flux  paths  is  very  nearly  halved,  and  the 
value  of  k  in  formulas  (72  and  (73)  should  therefore  be  doubled. 
If  it  is  desired  to  consider  the  increased  flux  due  to  the  use  of 
steel  binding  wires  or  bands,  this  can  be  done  by  making  a  suit- 
able correction  to  the  factor  k.1 

Should  it  be  desired  to  calculate  separately  the  average  value 
of  the  total  e.m.f.  generated  in  the  end  connections  of  the  short- 
circuited  coil,  we  have 


108 
where  le  =  total  length  of  coil  outside  armature  slots   (both 

ends)  in  centimeters, 
=  2*\/2\n,  if  we  keep  to  the  assumption  of  angle  a  = 

45  degrees, 
and  V  =  speed  of  cutting,  in  centimeters  per  second, 

=  —  7^,   in   which   D  is   the   armature   diameter   in 

V2 

centimeters,  and  N,  is  the  speed  in  revolutions  per  second. 
The  factor  \/2  is  the  necessary  correction  to  give  the  component 
of  the  velocity  at  right  angles  to  the  conductor.  Inserting  the 
value  of  Be  given  by  formula  (73),  and  making  the  required 
simplifications,  the  formula  for  voltage  becomes 

k  X  0.8V2nIcT*  frZW.)  (log,  2n  -  1) 


1  This  is  discussed  by  MR.  LAMME  in  his  Institution  paper  previously 
referred  to. 


160          PRINCIPLES  OF  ELECTRICAL  DESIGN 

The  numerical  value  of  k  in  multipolar  machines  of  modern 
design  will  usually  lie  between  1.3  and  3.5,  the  high  value  being 
taken  when  the  end  connections  lie  on  a  steel  or  iron  supporting 
cylinder  against  which  they  are  held  by  bands  of  steel  wire. 

For  first  approximations  the  formula  may  be  put  into  simpler 
form. 

Let  Ac  stand  for  the  ampere-conductors  per  pole  pitch  of  arma- 
ture periphery  (Ac  =  2TnIc).  Let  k  —  2.4  (being  an  average 
value),  and  for  the  quantity  (loge  2n  —  1)  put  the  numerical 
value  2.2  (the  assumption  here  being  that  there  are  12  to  14 
slots  per  pole),  then 

3TA  V 

Ee  (approximately)  =      1Q8C  (75) 

where  V  is,  as  before,  the  peripheral  velocity  of  the  armature 
in  centimeters  per  second. 

The  above  calculations  and  conclusions  are  based  on  the 
assumption  of  a  full-pitch  winding.  With  a  chorded  or  short- 
pitch  winding,  the  average  flux  density, in  the  commutating  zone 
will  be  slightly  reduced;  and  there  will  be  a  further  gain  due  to 
the  shortening  of  the  end  connections  (ABC  and  A'B'C'  in 
Fig.  59).  Thus  the  voltage  generated  by  the  cutting  of  the  end 
fluxes,  with  a  short  pitch  winding,  will  be  slightly  less  than  the 
value  calculated  by  formula  (74) ,  or  by  the  approximate  formula 
(75),  which  applies  to  a  full-pitch  winding. 

49.  Calculation  of  "Slot  Flux"  Cut  by  Coil  during  Commuta- 
tion.— A  reference  to  the  diagrams  of  flux  distribution  in  the 
commutating  zone  (Figs.  57  and  58)  will  make  clear  the  fact  that, 
even  when  the  effect  of  the  end  connections  is  neglected,  the 
center  of  the  neutral  commutating  zone  is  not  the  point  on  the 
armature  periphery  where  flux  neither  enters  nor  leaves  the  teeth ; 
because  in  order  that  the  short-circuited  conductors  shall  not 
cut  the  slot  leakage  flux,  this  flux  must  be  provided  by  the 
main  field  pole  toward  which  the  brushes  are  shifted  to  obtain  per- 
fect commutation.  The  point  on  the  armature  periphery  where 
flux  neither  enters  nor  leaves  the  teeth  may  be  found  by  drawing 
curves  representing  the  magnetomotive  forces  exerted  by  field 
poles  and  armature  windings  at  every  point  on  the  armature 
periphery,  and  where  the  sum  of  the  ordinates  of  such  curves  is 
zero  the  surface  flux  density  must  also  be  zero.  The  brushes 
must,  however,  be  moved  forward  beyond  this  point  until  the 
reversing  flux  entering  the  teeth  comprised  in  the  brush  arc  has 


'COMMUTATION  161 

the  value  3>e  as  given  by  the  formula  (72),  to  compensate  for  the 
end  fluxes,  plus  the  total  slot  flux  <£s,  which  is  twice  the  leakage 
from  tooth  to  tooth  in  one  slot  when  the  conductors  are  carrying 
the  full  armature  current.1  In  practice,  when  we  wish  to  cal- 
culate the  volts  generated  in  the  coil  of  T  turns  by  this  slot 
leakage  flux,  it  is  the  equivalent  slot  flux  that  must  be  considered, 
because  the  total  number  of  lines  crossing  between  the  sides  of 
adjacent  teeth  does  not  link  with  all  the  wires  in  the  coil. 

It  will  be  convenient  to  assume  the  same  number  of  slots 
as  there  are  commutator  bars,  and  the  whole  of  the  slot  space 
to  be  filled  with  2T  conductors,  each  carrying  a  current  of  Ic 
amp.  (this  follows  from  the  assumption  of  a  full-pitch  winding). 
Thus  no  account  will  be  taken  of  the  fact  that  a  small  space 
occurs  between  upper  and  lower  coils,  where  the  slot  flux  will 
not  pass  through  the  material  of  the  conductors.  The  lines  of 
the  slot  flux  will  be  supposed  to  take  the  shortest  path  from  tooth 
to  tooth;  the  small  amount  of  flux  that  may  follow  a  curved 
path  from  corner  to  corner  of  tooth  at  the  top  of  the  slot  will  be 
neglected.  Refinements  of  this  nature  may  be  introduced,  if 
desired,  when  solving  the  problem  for  a  concrete  case.2  If  the 
usual  assumption  is  made  that  the  reluctance  of  the  iron  in  the 
path  of  the  magnetic  lines  is  negligible  in  comparison  with  the 
slot  reluctance,  the  small  portion  of  slot  flux  in  the  space  dx 
(Fig.  62)  considered  1  cm.  long  axially  (i.e.,  in  a  direction  per- 
pendicular to  the  plane  of  the  paper)  is 

d$  =  m.m.f.  X  dP 
where  dP  is  the  permeance  of  the  air  path.     Thus, 

d$  =  0.47T  (277C)  -,  X  — 

CL  S 

1  In  the  case  of  short-pitch  windings,  or  when  there  are  more  commutator 
bars  than  there  are  slots  on  the  armature,  the  amount  of  the  slot  flux  must 
be  calculated  for  the  instant  when  the  coil  enters  or  leaves  the  commutating 
zone.     This  flux  will  depend  not  only  upon  the  dimensions  of  the  slot  but 
also  upon  the  current  carried  by  each  conductor  and  the  position  of  the 
latter  in  the  slot. 

2  With  short-pitch  windings,  or  when  there  are  several  coils  per  slot,  all 
the  conductors  in  the  slot  may  not  be  carrying  the  full  armature  current 
when  the  coil  enters  or  leaves  the  commutating  zone.     In  such  cases  the 
actual  conditions  must  be  studied,  an  average  value  for  the  slot  flux  being 
readily  arrived  at.     If  necessary,  the  straight-line  law  of  commutation 
may  be  assumed  in  order  to  estimate  the  current  values  in  the  coils  passing 
through  the  intermediate  stages  of  commutation. 

11 


162 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


but  this  flux  links  with  only  2T  X  --,  conductors,  and  the  equiva- 

Ci 

lent  slot  flux,  which  would  generate  the  same  e.m.f.  if  cut  by  all 
the  conductors  in  the  slot,  is  therefore 


(equivalent)     = 


X 

X    j 


and 


\J  •  \JI\    JL    J.  c 

*** '(equivalent)    =  ^2^ 

_  Q.Sird 
~      3s 


(76) 


FIG.  62. — Illustraing  slot-flux  calculations. 

This  is  the  equivalent  flux  in  maxwells  per  centimeter  of  axial 
length  of  armature  slot.  If  la  =  axial  length  of  armature  core 
expressed  in  centimeters,1  the  total  slot  flux  cut  by  each  coil-side 
during  commutation,  being  twice  the  flux  per  slot,  as  shown  in 
Fig.  57,  is 

*..^r/j.  .  (77) 

The  voltage  component  due  to  the  cutting  by  both  coil-sides2 
of  the  slot  flux  considered  separately  from  other  fluxes  would  be 


tc    = 


where  the  time  of  commutation 

brush  arc  in  centimeters  of  armature  periphery 
peripheral  velocity  in  centimeters  per  second 

Wa 

V 

Thus, 


3  X  10W  a 

The  factor  Wa  in  the  denominator  of  this  formula  indicates  that 
the  slot  flux  is  of  less  importance  with  a  wide  than  with  a  narrow 

1  It  is  well  to  let  la  stand  for  the  gross  length  of  armature  core,  although 
in  slot  flux  calculations  the  total  width  of  vent  ducts  is  sometimes  deducted. 

2  The  2,T  conductors  in  the  one  slot  are  here  considered  as  equivalent  to 
the  two  coil-sides,  each  of  T  wires,  in  separate  slots  one  pole  pitch  apart. 


COMMUTATION  163 

brush.  This  is  generally  true,  although  it  must  be  remembered 
that  the  formula  (78),  in  common  with  other  formulas  pre- 
viously derived,  is  not  of  general  application.  Within  the  limits 
of  this  chapter,  it  is  not  possible  to  consider  all  special  cases; 
and  commutation  formulas  of  general  applicability  cannot  be 
developed.  When  there  is  more  than  one  coil  per  slot,  and  when 
there  are  "dead  coils,"  inequalities  occur  which  complicate  the 
problem  and  make  it  impossible  to  obtain  ideal  commutation 
with  every  coil  on  the  armature.  In  such  cases  the  slot  flux  must 
be  calculated  for  the  coils  that  are  differently  situated  in  regard 
to  the  brush  position  and  an  average  value  selected  for  use  in  the 
calculations. 

Knowing  the  slot  flux  $e»  and  the  previously  calculated  end  flux 
$«  cut  by  the  conductors  of  the  short-circuited  coil  while  travel- 
ling over  the  distance  Wa,  the  correct  brush  position  is  found  when 
the  reversing  flux  entering  the  teeth  comprised  in  the  commu- 
tating  zone  of  width  Wa,  is  approximately  3>c,  +  $e  maxwells. 
The  flux  actually  cut  by  the  one  coil-side  is,  however,  only  $e; 
the  component  3?e,  of  the  total  flux  entering  the  commutating 
zone  merely  supplies  the  leakage  from  tooth  to  tooth  across  the 
slot.  The  presence  of  the  slot  flux  undoubtedly  tends  to  com- 
plicate the  problem  of  commutation.  It  should  be  noted  that 
the  slot  flux  $eg,  if  calculated  by  formula  (77),  is  what  has  been 
referred  to  as  the  equivalent  slot  flux;  that  is  to  say,  a  flux  of 
this  value,  if  cut  by  an  imaginary  concentrated  winding  of  T 
turns,  would  develop  the  same  voltage  in  the  coil  as  the  actual 
slot  flux  develops  in  the  actual  winding.  The  condition  of 
importance  to  be  fulfilled  is  simply  that  the  "equivalent"  flux 
cut  by  the  coil-side  in  the  reversing  field  shall  have  the  value  $<,. 
The  flux  cut  by  the  coil-side  may  be  separated  into  two  parts:  (1) 
the  flux  passing  through  the  teeth  into  the  armature,  which  links 
with  all  the  conductors  in  the  slot,  and  (2)  the  equivalent  slot 
flux.  It  is  important  to  note  that  although  the  total  or  actual 
slot  flux  is  the  same  whether  it  enters  the  top  or  the  root  of  the 
tooth,  the  flux  linkage  and  therefore  the  developed  voltage  have 
not  the  same  value  in  the  two  cases.  The  total  slot  flux,  on  the 
basis  of  the  assumptions  previously  made,  is 


= 


as 


Jo 


164          PRINCIPLES  OF  ELECTRICAL  DESIGN 

being  one  and  one-half  times  the  equivalent  slot  flux  given  by 
formula  (77).  The  equivalent  flux  when  the  total  slot  flux  enters 
the  tooth  top  instead  of  passing  through  root  of  tooth  is  no  longer 
expressed  by  formula  (77)  ;  it  may  be  calculated  thus  : 

The    magnetic    lines    represented    by    the   expression  d$  = 

f  fJ^r  i* 

0.4?r  (2TIC)  -j  —  no  longer  link  with  2T  -,  conductors,  but  with 
d  s  ct 

2T-  —  -j—  -  conductors  (see  Fig.  62).     The  equivalent  flux,  when 

no  part  of  this  flux  passes  into  the  armature  core  below  the  teeth, 
is  therefore 


x 


I 


d 

x(d  —  x)dx 


-^HA  (80) 

or  just  half  the  equivalent  slot  flux  as  given  by  formula  (77). 
The  question  now  arises :  What  is  the  necessary  total  flux  enter- 
ing the  tops  of  the  teeth  comprised  in  the  commutating  zone  to 
develop  the  proper  voltage  component  in  the  short-circuited 
coil? 

A  total  slot  flux  as  given  by  formula  (79)  has  the  "equivalent" 
value  as  given  by  formula  (80) ;  that  is  to  say,  it  is — on  the  basis 
of  the  assumptions  previously  made — three  times  as  great  as 
the  equivalent  flux.  The  total  flux  entering  the  teeth  comprised 
in  the  commutating  zone  should  therefore  be 

3>c  =  3<£'eS  +  flux  passing  directly  into  armature  core  through 
the  teeth. 

but  $'es  +  $d  =  &e,  where  3>e  is,  in  this  particular  instance,  the 
equivalent  value  of  the  total  flux  to  be  cut  by  the  "active" 
portion  of  the  short-circuited  conductors. 
Thus 

^Ac     '          ^Mr    £s       I        JT Q  ^O-Ly 

or,  if  preferred, 

*c    =  *„   +   *.  (82) 

where  $e8  is  the  equivalent  slot  flux  as  originally  calculated  and 
expressed  by  formula  (77). 


COMMUTATION  165 

The  flux  actually  entering  the  armature  teeth  in  the  com- 
mutation zone  should  therefore  be  equal  to  the  sum  of  the  end 
flux  $e  and  the  equivalent  slot  flux  $es.  It  is  because  this  con- 
clusion is  not  obvious  that  it  has  been  deduced  from  the  fore- 
going arguments. 

Having  determined  the  value  of  the  flux  $c  which  must  enter 
the  teeth  comprised  in  the  commutating  zone  of  width  Wa, 
it  is  evident  that  the  average  air-gap  density  in  this  zone,  to 
produce  perfect  commutation,  must  be 

*'-wr*i.  (83) 

By  referring  to  the  final  flux  distribution  curve,  C,  obtained  by 
the  method  outlined  in  Art.  43,  Chap.  VII,  it  may  easily  be 
seen  whether  or  not  the  desired  field  can  be  obtained  in  the 
fringe  of  the  leading  pole  tip.  If  the  required  field  is  greater  than 
that  obtainable  in  the  interpolar  space,  commutating  poles  must 
be  provided,  or  the  machine  must  be  re-designed.  If  the  flux- 
distribution  curves  have  not  been  drawn,  the  calculated  density 
Bc  required  in  the  commutating  zone,  as  expressed  by  formula 
(83),  may  be  compared  with  the  average  air-gap  density  under 
the  main  poles.  If  the  required  density  does  not  exceed  one-half 
of  the  average  density  of  the  main  flux  taken  over  the  pole  pitch, 
it  will  usually  be  possible  to  obtain  satisfactory  commutation  in 
the  fringe  of  the  leading  pole  tip,  provided  carbon  brushes  are 
used.  In  the  event  of  Bc  being  in  excess  of  this  value,  interpoles 
will  probably  be  necessary. 

50.  Commutating  Interpoles. — Assuming  the  same  number  of 
interpoles  as  there  are  main  poles,  and  an  axial  length  of  interpole 
equal  to  that  of  the  main  pole,  the  flux  from  each  interpole 
which  enters  the  armature  teeth  included  in  the  commutating 
zone  of  width  Wa  is,  as  before,  $ea  +  $e. 

If,  as  is  usually  the  case,  the  interpole  face  does  not  cover 
the  whole  length  of  the  armature  core,  then  some  flux  due  to  the 
total  m.m.f.  of  the  armature  windings  will  enter  or  leave  the 
armature  core  by  the  teeth  included  in  the  commutating  zone, 
and  this  flux  will  be  cut  by  that  portion  of  the  slot  conductors 
which  is  not  covered  by  the  interpole.  With  the  brushes  on 
the  geometric  neutral  line,  this  armature  flux  is  unaffected  by 
the  excitation  of  the  main  poles ;  its  value  depends  only  upon  the 
armature  ampere-turns  and  the  reluctance  of  the  air  paths 


166 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


between  the  armature  surface  and  neighboring  masses  of  iron. 
It  can  be  predetermined  within  reasonably  close  limits  by  plot- 
ting the  full-load  flux  curve,  C,  as  indicated  in  Art.  43,  Chap. 
VII.  The  flux  entering  and  leaving  the  surface  of  the  armature 
in  the  commutating  zone,  when  the  axial  length  lp  of  the  interpole 
is  appreciably  less  than  the  gross  length  la  of  the  armature  core, 
is  indicated  in  Fig.  63.  Before  calculating  the  flux  which  must 
enter  the  armature  teeth  from  the  commutating  pole,  it  will 
be  advisable  to  define  clearly  the  various  flux  components  to  be 
considered. 


FIG.  63. — Commutating  pole :  showing  direction  of  flux  at  armature  surface. 

Many  of  the  symbols  used  in  the  previous  calculations  will  be 
employed,  but  it  is  proposed  to  alter  the  meaning  of  some  of 
these  because  it  will  be  more  convenient  to  think  of  the  slot 
flux  per  centimeter  length  of  slot  instead  of  the  flux  over  the 
whole  length  of  slot  as  in  the  previously  developed  formulas. 
This  slight  change  will  probably  lead  to  less  confusion  than  if 
a  complete  new  set  of  symbols  were  to  be  introduced  here. 

<£c  =  total  flux  entering  armature  teeth  from  interpole,  over 

area  of  width  Wa  and  length  lp. 
&e  =  total  end  flux  (one  end  of  armature). 
$>,  =  total  slot  flux  per  centimeter  of  armature  length  (two 

slots). 
&e8  =  equivalent  slot  flux  per  centimeter,  if  magnetic  lines 

pass  outward  from  armature  core  through  root  of  teeth 

(two  slots). 
$'es  =  equivalent  slot  flux  per  centimeter,  if  magnetic  lines  pass 

inward  from  air  gap  through  top  of  teeth  (two  slots). 


COMMUTATION  167 

<l>d  =  portion  of  interpole  flux  per  centimeter  length,  which 
enters  armature  core  through  root  of  teeth. 

$a  =  armature  flux  per  centimeter  length,  which  leaves 
teeth  over  the  commutating  zone  of  width  Wa  and  length 
la  -  IP  (Fig.  63). 

The  equivalent  flux  to  be  cut  by  conductors  under  the  interpole 
must  equal  the  total  of  all  the  flux  components  that  have  to  be 
neutralized.  This  leads  to  the  equation 

$dlp  +  Veslp    =$e+  *a(la    ~   lp)    +  $>es(la   ~   lp) 

from  which  a  value  for  <f>d  can  be  calculated.  The  total  flux 
leaving  interpole  is 

*c  =  $slp  +  $>dlp 

Inserting  for  $<*  in  this  last  equation  the  value  derived  from  the 
previous  equation,  we  get 

$c  =  $.lp  +  <*>,+  $>a(la  -  IP)  +  Mk  -  IP)  -  *'.Jp          (84) 

This  equation  may  be  simplified  by  expressing  to  total  slot  flux 
$,  and  the  equivalent  slot  flux  $'M  in  terms  of  the  equivalent  slot 
flux  $e»-  The  relation  between  these  quantities  is  obtained  by 
comparing  the  previously  developed  equations  (79),  (80)  and 

(77).     Thus, 

$>,    =    %  *e, 

and 

*'..    =    1A  *e, 

Inserting  these  values  in  equation  (84)  we  get 

$e    =   *.  +  $e,  la   +  ^a    ~    IP)  (85) 

wherein  the  symbols  &e,  and  $a  stand  for  flux  components  per  unit 
length  of  armature  core,  as  previously  mentioned. 

Knowing  the  amount  of  the  flux  to  be  provided  by  each  inter- 
pole, its  cross-section  can  be  decided  upon  and  the  necessary 
exciting  ampere-turns  calculated,  bearing  in  mind  the  following 
requirements : 

(a)  The  average  air-gap  density  should  be  low  (about  6,000 
gausses  corresponding  to  full-load  current),  to  allow  of  increase 
on  overloads. 

(b)  The  leakage  factor  should  be  as  small  as  possible.     This 
involves  keeping  the  width  and  axial  length  of  interpole  small, 


168          PRINCIPLES  OF  ELECTRICAL  DESIGN 

thus  conflicting  with  condition  (a)  and  presenting  one  of  the 
difficulties  of  commutating-pole  design. 

(c)  the  minimum  width  of  pole  face  must  be  such  that  the  equi- 
valent pole  arc  (which  must  include  an  allowance  for  fringing) 
shall  cover  the  commutating  zone  of  width  Wa. 

(d)  The  equivalent  pole  arc  should,  if  possible,  be  an  exact 
multiple  of  the  slot  pitch  (either  once  or  twice  the  slot  pitch) 
as  this  tends  to  reduce  the  magnitude  of  the  flux  pulsations  in 
the  interpole.     The  effects  of  these  flux  pulsations,  caused  by 
variations  in  the  reluctance  of  the  interpole  air  gap,  are,  however, 
usually  of  no  great  practical  importance,  but  the  width  of  brush 
should  not  be  determined  independently  of  the  interpole  design. 

(e)  In  order  to  keep  down  the  PR  losses  in  the  series  turns  on 
the  interpole  (usually  amounting  to  less  than  1  per  cent,  of  the 
total  output),  the  ampere- turns  and  the  length  per  turn  should 
be  as  small  as  possible.     The  gain  resulting  from  a  small  air 
gap  is,  however,  not  great,  because  the  ampere-turns  required 
to  overcome  the  air-gap  reluctance  rarely  exceed  25  per  cent, 
of  the  total,  the  balance  being  required  to  oppose  the  armature 
m.m.f.     A  reasonably  large  air  gap  has  the  advantage  of  re- 
ducing the  flux  pulsations  referred  to  under  (d). 

(f)  The  effect  of  the  interpole  being  to  increase  the  flux  in 
that  portion  of  the  yoke  which  lies  between  the  interpole  and 
the  main  pole  of  opposite  polarity,  it  is  important  to  see  that 
the  resulting  flux  density  in  this  part  of  the  magnetic  circuit  is 
not  excessive.     A  similar  condition  exists  in  the  armature  core, 
but  this  does  not  usually  determine  the  limits  of  the  allowable 
average  flux  density  below  the  teeth. 

(g)  Series-  or  wave-wound  armatures  are  to  be  preferred  on 
machines  with  commutating  poles,  especially  when  the  air  gap 
under  the  main  poles  is  made  smaller  than  it  would  have  to  be 
if  interpoles  were  not  used. 

(h)  The  total  line  current  should,  if  possible,  pass  through 
all  the  interpole  windings  in  series;  that  is  to  say,  parallel 
circuits  should  be  avoided  because  of  the  possibility  that  the 
current  may  not  be  equally  divided.  If  the  total  current  is 
too  great,  a  portion  may  be  shunted  through  a  diverter.1  The 
diverter  should  be  partly  inductive,  the  resistance  being  wound 

1  The  use  of  a  resistance  as  a  shunt  to  the  series  field  winding — usually 
known  as  a  diverter — will  be  referred  to  again  later  when  considering  the 
design  of  the  field  magnets. 


COMMUTATION  169 

on  an  iron  core  in  order  that  the  time  constants  of  the  main  and 
shunt  circuits  may  be  approximately  equal.  If  this  is  not  done, 
the  interpole  winding  will  not  take  its  proper  share  of  the  total 
current  when  the  change  of  load  is  sudden,  and  this  may  lead  to 
momentary  destructive  sparking. 

Among  the  advantages  of  commutating  poles  may  be  men- 
tioned the  fixed  position  of  the  brushes  and  the  fact  that  fairly 
heavy  overloads  can  be  taken  from  the  machine  without  de- 
structive sparking,  because  of  the  building  up  of  the  commutat- 
ing flux  with  increase  of  load.  The  limiting  factor  in  this  con- 
nection is  the  saturation  of  the  iron  (mainly  of  the  interpole 
itself)  in  the  local  circuit,  and  this  is  aggravated  by  the  large 
percentage  of  leakage  flux  due  to  the  proximity  of  main  and  com- 
mutating poles.  Deeper  armature  slots  may  be  used  than  in 
the  case  of  machines  without  interpoles,  and  the  specific  load- 
ing (ampere-conductors  per  unit  length  of  armature  periphery) 
may  be  increased,  thus  allowing  of  greater  output  notwithstand- 
ing the  slight  reduction  in  width  of  main  poles  necessary  to 
accommodate  the  interpoles.  The  maximum  output  of  the 
machine  with  commutating  poles  is  usually  determined  by  the 
heating  limits,  the  ventilation  being  less  effective  than  in  the 
case  of  non-interpole  machines.  The  PR  loss  in  interpole 
windings  is  to  some  extent  compensated  for  by  a  reduction  of 
the  ampere-turns  on  the  main  poles  when  shorter  air  gaps  are 
used.  With  the  brushes  on  the  geometric  neutral  and  an  air 
gap  which  is  small  relatively  to  the  space  between  pole  tips, 
field  distortion  per  se  has  nothing  to  do  with  commutation, 
whether  interpoles  are  used  or  not;  if  the  fringe  from  the  leading 
pole  tip  is  not  to  be  used  for  counteracting  the  effects  of  end  flux 
and  slot  flux  on  the  coil  undergoing  commutation,  the  unequal 
flux  distribution  under  the  main  poles  due  to  cross-magnetization 
does  not  affect  the  field  at  a  point  midway  between  two  main 
poles.  It  is  not  suggested  that  field  distortion  in  unobjectionable 
when  the  brushes  are  on  the  geometric  neutral  or  when  inter- 
poles are  used.  The  concentration  of  flux  at  one  side  of  the  main 
pole  may  lead  to  flashing  over  the  commutator  surface  (an  effect 
often  attributed  to  unsatisfactory  commutation,  though  rarely 
due  to  this  cause);  but  the  chief  objection  to  a  large  number  of 
armature  ampere-turns  per  pole  is  the  fact  that  the  flux  in  the 
zone  of  commutation  due  to  this  m.m.f.  must  be  compensated  for 
somehow  if  satisfactory  commutation  is  to  be  obtained.  It 


170 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


is  exactly  in  the  zone  corresponding  to  the  brush  position  that 
the  armature  m.m.f.  has  its  maximum  value.  In  the  case  of 
the  interpole  machine,  the  windings  necessary  to  compensate 
for  armature  cross-magnetization  are  an  objectionable  feature, 
and,  except  for  the  added  cost  and  tendency  to  interfere  with 
ventilation,  there  is  much  to  be  said  in  favor  of  pole-face  wind- 
ings, the  function  of  which  is  to  neutralize  the  magnetizing 
effect  of  the  armature  winding  and  maintain  an  approximately 
constant  flux  density  over  the  pole  faces.  The  writer  has  in 
mind  machines  such  as  those  which,  for  the  last  20  years,  have 
been  constructed  under  the  THOMPSON-RYAN  patents.  One 


\ 


Center  Line  of  Pole 


Compensating  Coi 
Main  Exciting  Coi] 
lutorpole  Winding 


FIG.  64. — Compensating  pole-face  winding. 

of  the  attractive  features  of  such  designs  is  the  fact  that  the 
winding  on  the  commutating  poles  need  be  no  greater  than  that 
required  to  overcome  the  reluctance  of  the  air  gap  and  send  the 
requisite  flux  into  the  armature  teeth  comprised  in  the  commu- 
tating zone.  A  pole-face  compensating  winding  is  shown  in 
Fig.  64.  The  balancing  coils  pass  through  slots  in  the  pole 
face  and  carry  the  full  current  of  the  machine;  that  is  to  say, 
they  are  connected  in  series  with  the  commutating-pole  windings 
and  the  compounding  series  turns  (if  any)  on  the  main  poles. 
The  connections  between  the  pole-face  compensating  coils  are 
so  made  that  the  current  in  these  will  always  tend  to  neutralize 
the  magnetic  effect  of  the  currents  in  the  armature  coils,  and  so 
prevent  distortion  of  the  flux  over  the  pole  face. 


COMMUTATION  171 

51.  Example  of  Interpole  Design.  —  The  numbers  and  dimen- 
sions used  to  illustrate  the  method  of  calculation  outlined  above 
will  be  chosen  without  reference  to  an  actual  design  of  interpole 
dynamo,  and  they  must  not  be  considered  representative  of 
modern  practice.  Assume  the  leading  particulars  of  the  machine 
to  be  as  follows: 

Output  =  200  kw. 

Volts  =  440. 

R.p.m.  =  500. 

Number  of  main  poles  p  =  6. 

Armature  core  diameter  D  =  30  in. 

Armature  core  length  la  =  11  in.  =  28  cm. 

Total  number  of  slots  =  120. 

Number  of  slots  per  pole  n  =  20. 

Slot  pitch  X  =  0.785  in. 

Slot  width  s  =  0.39  in. 

Slot  depth  d  =  1.5  in. 

Style  of  winding:  full-pitch,  multiple. 

Current  per  armature  path  Ic  —  76  amp. 

Number  of  conductors  per  slot  =  8. 

Total  number  of  conductors  Z  =  120  X  8  =  960. 

Number  of  commutator  bars  =  240.  (There  are  four  coil- 
sides  per  slot,  or  two  coils,  giving  two  turns  between  adjacent 
commutator  bars.) 

Diameter  of  commutator  =  20  in. 

Pitch  of  commutator  bars  =  ^940     =  O-2^2  in. 

Number  of  bars  covered  by  brush  =  3.5. 

Thickness  of  brush  (brush  arc)  W  =  0.262  X  3.5  =  0.916  in. 

Brush  arc  referred  to  armature  periphery 

Wm  , 


Assuming  the  same  number  of  commutating  poles  as  there  are 
main  poles,  the  flux  entering  the  armature  teeth  in  the  commutat- 
ing zone  of  width  Wa  should  have  the  value  given  by  formula  (85)  . 
The  end  flux  cut  by  the  short-circuited  coils  is  given  approxi- 
mately by  formula  (72)  in  which  the  coefficient  k  may  be  given 
the  value  2.  Thus: 

3>e  =  0.4V2fcr/cnTF.[(lo&  2n)  -  1] 

=  0.4\/2  X  2  X  4  X  76  X  20  X  3.5  X  (3.69  -  1) 
=  64,800  maxwells. 


172 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


In  regard  to  the  slot  flux,  at  the  beginning  or  end  of  commuta- 
tion (depending  upon  the  position  of  the  coil  in  the  slot)  the  con- 
ductors in  the  slot  affected  are  not  all  carrying  the  full  armature 
current.  This  will  be  seen  by  reference  to  Fig.  65,  where  coil 
A  (consisting  of  two  turns)  is  just  about  to  be  short-circuited  by 
the  brush.  At  this  instant  the  current  in  the  coil-sides  B  and  D 

is  i  =  27C  X  5-?  =  0.572/c  as  indicated  by  the  diagram  sketched 

o.O 

on  the  brush  of  width  W  covering  three  and  one-half  bars,  the 


.  

1 
1 

j 

! 

i 
1 

«—  "i 

fft  Q  ~  "~  7 

j    /\ 4  t 


FIG.  65. — Illustrating  example  of  interpole  design. 


straight-line  law  of  current  variation  being  assumed.  At  the  end 
of  commutation,  when  the  position  of  the  affected  slot  relatively 
to  brush  and  interpole  is  as  indicated  by  the  dotted  lines,  the 
current  in  all  four  coil-sides  is  —  7C.  We  may,  therefore,  assume 
the  slot  flux  to  be  produced  by  a  current  of  which  the  average 

3  +  0  572 
value  is  not  Ic  =  76  amp.,  but  Ic  X ~r~   -  =  68  amp.     The 

slot  flux  $es  in  formula  (85),  which  is  the  " equivalent"  value  as- 


COMMUTATION 


173 


suming  the  total  slot  flux  to  pass  outward  from  the  armature  core 
through  the  teeth,  is,  therefore,  by  formula  (77), 

l&vd 


whence, 


TIe 


1.6*  X  1.5X4X68X11  X  2.54 


3  X  0.39 
=  49,000  maxwells, 

this  being  the  equivalent  slot  flux  taken  over  the  full  length  of 
the  armature. 

Turning  now  to  the  flux  leaving  the  armature  in  that  part  of 
the  commutating  belt  which  is  not  covered  by  the  interpole,  let 
the  curve  Fig.  66  represent  the  full-load  flux  distribution  over 


FIG.  66. — Flux  curve  showing  density  on  geometric  neutral  line, 
lated  without  interpoles.) 


(Calcu- 


armature  surface,  calculated  on  the  assumption  that  there  are 
no  interpoles.  This  is  the  flux  curve  C,  derived  in  the  manner 
outlined  in  Art.  43,  Chap.  VII,  but  with  the  brushes  left  in  the 
no-load  position.  There  will  be  no  directly  demagnetizing  effect 
due  to  brush  lead  since  the  brushes  will  remain  on  the  geometric 
neutral  line.  The  flux  density  at  a  point  midway  between  the 
two  poles  has  the  value  B,  which  for  the  purpose  of  this  example 
may  be  assumed  to  be  1,500  gausses. 

If  lp  =  axial  length  of  interpole  (not  yet  determined),  the  total 
flux  which  must  pass  from  interpole  into  armature  teeth  is 

$c    =   $e   +  $Ja  +  BWa(la  ~  lp) 


174          PRINCIPLES  OF  ELECTRICAL  DESIGN 

which  is  simply  formula  (85)  with  the  armature  flux  per  centi- 
meter of  length  expressed  as  BWa  instead  of  $a-  The  axial 
length  lp  of  the  interpole  face  can,  therefore,  be  determined  if  a 
suitable  value  for  the  average  air-gap  density  under  full-load  con- 
ditions is  assumed.  Let  Bp  stand  for  this  value;  then 

BplpWa    =   Qe^esla  +  BWa(la   ~    lp) 

whence 

lp  =  "  e  Wa(Bp+B)  °  (86) 

If  the  flux  densities  are  expressed  in  gausses,  the  dimensions  must 
be  in  centimeters,  and  if  Bp  is  taken  as  4,500  gausses,  the  length 
lp  is  found,  by  formula  (86),  to  be  12.4  cm.  or,  say,  5  in. 
The  total  flux  in  the  interpole  air  gap  at  full  load  is  then, 

$c  =   64,800  +  49,000  +  1,500  X  6.45  X  1.375(11  -  5) 
=  193,600  maxwells. 

Assuming  a  leakage  factor  of  1.9  and  a  cross-section  under 
interpole  winding  of  (5  X  1J^)  sq.  in.,  the  full-load  density  in  the 
core  of  the  interpole  would  be  7,600  gausses. 

Calculation  of  Ampere-turns  Required  on  Interpole. — Referring 
to  Fig.  65,  it  will  be  seen  that  when  the  coil  A  is  about  to  be  short- 
circuited,  the  interpole  flux  enters  the  armature  through  the 
teeth  1  and  2.  At  the  end  of  commutation  this  flux  enters  the 
teeth  3  and  4.  The  permanence  of  the  air  gap  of  length  d  may 
vary  slightly  with  the  change  in  the  position  of  the  armature 
teeth;  it  may  be  calculated  by  any  of  the  approximate  methods.1 
Assuming  an  actual  air  gap  J4  in-  long,  the  equivalent  air  gap 
might  be  0.3  in.  The  full-load  ampere-turns  required  to  over- 
come air-gap  reluctance  will  therefore  be 
0.3  X  2.54  X  4,500 

f\  A  —         )  i  £\J 

To  this  must  be  added  the  ampere-turns  to  oppose  the  armature 
m.m.f.  If  the  reduction  of  current  in  the  short-circuited  coils  is 
neglected,  the  armature  ampere-turns  per  pole  are 

IZ/c  =  (120  X  8)  X  76 

2  p  2X6 

=  6,080 

If  we  neglect  the  very  small  m.m.f.  required  to  overcome  the  re- 
luctance of  the  interpole  core,  the  total  ampere-turns  on  each 
1  See  Art.  36,  p.  115. 


COMMUTATION 


175 


interpole  should  be  2,720  +  6,080  =  8,800  at  full  load.  The 
full-load  current  of  the  machine  is  200,000  -f-  440  =  455  amp., 
and  the  required  number  of  turns  is  8,800  4-  455  =  19.35.  In 
practice  about  21  turns  would  be  put  on  the  interpole  of  this 
machine,  and  if  necessary  a  diverter  would  be  provided  to  adjust 
the  current  in  accordance  with  results  obtained  on  test. 

52.  Prevention  of  Sparking — Practical  Considerations. — It  is 
not  suggested  that  the  method  of  considering  commutation  phe- 
nomena as  outlined  above  covers  the  subject  completely.  The 
designer  aims  at  obtaining  " ideal"  commutation  under  certain 
load  conditions,  knowing  well  that,  even  when  series-wound  corn- 
mutating  poles  are  used,  the  required  conditions  cannot  be  ex- 
actly fulfilled  at  other  loads.  He  relies  on  the  resistance  of  the 


FIG.  67. — Armature  coil  near  end  of  commutation  period. 

carbon  brush  to  give  sparkless  commutation  even  when  the  con- 
ditions depart  appreciably  from  those  of  "ideal"  commutation. 
The  extent  to  which  the  ideal  condition  can  be  departed  from 
without  producing  destructive  sparking  is  not  easily  determined 
except  by  experimental  means. 

In  Art.  45,  page  146  the  effect  of  the  brush-contact  resistance 
was  considered,  and  it  was  seen  that  the  value  of  this  resistance 
has  no  effect  on  the  problem  of  commutation  provided  the  change 
of  current  in  the  short-circuited  coil  takes  place  in  accordance 
with  the  "ideal"  or  straight-line  law.  The  reason  is  that,  when 
"  straight-line "  commutation  is  obtained,  the  distribution  of  the 
current  over  the  contact  surface  of  a  brush  of  rectangular  section 
is  necessarily  uniform.  If,  now,  we  wish  to  examine  the  condi- 
tions of  commutation  when  the  changes  of  current  do  not  follow 
the  ideal  straight-line  law,  it  is  necessary  to  consider  the  effect  of 
the  brush-contact  resistance  when  the  current  density  is  no  longer 
uniform^over  the  entire  surface.  The  diagram,  Fig.  67,  is  gener- 


176          PRINCIPLES  OF  ELECTRICAL  DESIGN 

ally  similar  to  Fig.  56,  except  that  the  coil  connecting  segments 
A  and  B  has  moved  nearer  to  the  edge  of  the  commutation  zone. 
The  distance,  in  inches,  still  to  be  travelled  before  the  removal  of 
the  short-circuit  is  w,  which  is  supposed  to  be  only  a  small  per- 
centage of  the  total  brush  arc  W. 

Let  R  be  the  resistance,  in  ohms,  of  the  coil  connecting  the 
commutator  segments  A  and  B. 

Rc  =  the  contact  resistance  of  the  brushes  per  square  inch  of 
area. 

lc  =  the  total  length,  in  inches,  of  the  set  of  brushes  measured 
parallel  to  the  axis  of  rotation. 

A  =  the  average  current  density,  in  amperes  per  square  inch, 
over  the  brush-contact  surface.  It  will  also  be,  approxi- 
mately, the  density  over  the  surface  Sb  of  the  segment 
B  (Fig.  67). 

AU,  =  the  maximum  permissible  current  density  over  the  sur- 
face of  the  brush  tip  of  width  w. 

Summing  up  the  e.m.fs.  and  potential  differences  in  the  path 
of  the  short-circuit  (the  resistance  of  the  material  in  the  body  of 
the  brush  being  neglected),  we  can  write,  for  the  value  of  the 
volts  developed  in  the  short-circuited  coil  at  the  instant  con- 
sidered, 

e  =  iR  +  A#c  -  &WRC 
But 

i  =  Ic  —  ia 

wherein  the  meaning  of  the  symbols  will  be  evident  from  an  in- 
spection of  Fig.  67.  This  last  equation  may  be  written, 

i  =  Ic  —  &wlcw 
Substituting  in  the  previous  equation,  we  get, 

e  =  ICR  -  &JewR  -  Rc(&w  -  A) 

If,  now,  we  imagine  w  to  become  smaller  and  smaller  as  it  ap- 
proaches zero  value,  the  second  term  on  the  right-hand  side  of 
the  equation  becomes  of  relatively  less  and  less  importance,  and 
we  may  therefore  write, 

e  =  ICR  -  Rc(kw  -  A) 

which  gives  an  approximate  value  for  the  permissible  e.m.f.  in 
the  short-circuited  coil  at  the  end  of  the  commutation  period. 
If  preferred,  this  equation  may  be  put  in  the  form 

e  =  ICR  -  Rc&(k  -  1)  (87) 


COMMUTATION  177 

wherein  k  is  the  ratio  of  the  permissible  density  at  brush  tip  to 
the  average  density  over  brush-contact  surface. 

For  values  of  A  above  30  amp.  to  the  square  inch,  the  voltage 
drop  R<A,  with  carbon  brushes,  is  usually  about  1  volt.  It  fol- 
lows that,  if  the  value  of  k  may  be  as  high  as  2.5,  the  actual  e.m.f . 
in  the  short-circuited  coil  may  differ  from  the  ideal  e.m.f.  by 
about  1.5  volts.  This  is,  however,  a  case  for  experimental 
determination ;  but  once  a  safe  value  for  k —  or  for  Aw — has  been 
determined,  the  allowable  variation  in  the  commutating  flux 
$c —  as  given  by  formulas  82,  page  164,  and  85,  page  167 — may 
readily  be  calculated.  If  the  assumed  value  of  1.5  volts  varia- 
tion is  permissible,  it  follows  that  the  amount  by  which  the  flux 
entering  the  teeth  in  the  commutating  zone  may  differ  from  the 
ideal  value  is 

^      1.5JCX108 

*=      ~^TT 

3le  X  108 
=  2  — JP —  maxwells  (88) 

4        L  c 

wherein  tc  is  the  time  of  commutation  in  seconds,  and  Tc  is  the 
number  of  turns  in  the  short-circuited  coil. 

Apart  from  all  considerations  of  a  mechanical  nature,  commu- 
tation can  be  improved  by  increasing  the  thickness  of  insulation 
between  commutator  bars.  This  might  in  many  cases  be  made 
considerably  thicker  than  the  usual  ^2  m-  with  advantage  in 
the  matter  of  sparking;  but  it  is  not  always  easy  to  obtain  large 
spacing  between  bars,  and  thick  mica  insulation  is  otherwise 
objectionable. 

When  calculating  the  equivalent  slot  flux,  the  assumption 
made  virtually  supposes  the  slot  to  contain  a  large  number  of 
small  wires  all  connected  in  series.  With  solid  conductors  of 
large  cross-section,  the  local  currents  in  the  copper  would  alter 
the  distribution  of  the  slot  flux  and  call  for  a  reversing  field 
differing  slightly  from  the  field  calculated  by  the  aid  of  the 
formulas  given  in  this  chapter.  Again,  the  field  due  to  the 
armature  m.m.f.  is  usually  assumed  to  be  stationary  in  space. 
This  is  practically  true  when  the  number  of  teeth  is  large  and 
the  brush  arc  is  a  multiple  of  the  bar  pitch.  With  few  teeth 
and  a  brush  covering  a  fractional  number  of  bars,  the  oscilla- 
tions of  the  armature  field  (of  small  magnitude  but  high  fre- 
quency) may  have  some  slight  effect  on  commutation;  but  with 
12 


178          PRINCIPLES  OF  ELECTRICAL  DESIGN 

a  better  understanding  of  the  main  principles  underlying  commu- 
tation phenomena  these  and  similar  modifying  factors  of 
secondary  importance  tend  to  assume  a  less  formidable  aspect. 
The  designer,  who  must  of  necesssity  be  an  engineer,  desires 
to  see  clearly  what  he  is  doing.  If  he  uses  formulas  of  which 
he  does  not  know  the  derivation  or  physical  significance,  he  is 
working  in  the  dark.  In  general,  he  asks  for  more  physics  and 
less  mathematics.  If  he  can  picture  the  short-circuited  coil 
cutting  through  the  various  components  of  the  flux  in  the 
commutating  zone,  and  understand  how  these  flux  components 
may  be  calculated  within  limits  of  accuracy  that  are  generally 
satisfactory  in  practice,  he  will  have  a  working  knowledge  of 
the  phenomena  of  commutation  which  should  be  especially 
valuable  in  cases  where  test  data  cannot  be  relied  upon. 

53.  Mechanical  Details  Affecting  Commutation. — The  quality 
of  the  carbon  used  for  the  brushes,  together  with  the  pressure 
between  brush  and  commutator  surface,  will  determine  the  heat- 
ing due  to  friction,  and  therefore,  to  some  extent,  the  dimensions 
and  proportions  of  the  commutator.  The  pressure  between 
brush  and  commutator  is  usually  adjusted  by  springs  so  that  it 
shall  be  from  1  to  2  Ib.  per  square  inch  of  contact  surface.  In 
order  to  avoid  excessive  temperature  rise,  the  current  density 
is  rarely  allowed  to  exceed  30  or  40  amp.  per  square  inch  of 
brush-contact  surface.  A  sufficient  cooling  surface  is  thus 
provided  from  which  the  heat  developed  through  friction  and 
PR  loss  may  be  radiated.  The  width  of  brush  (brush  arc) 
should  lie  within  the  limits  of  1J4  and  3^  commutator  bars; 
and  as  a  further  check  on  the  desirable  dimensions,  the  width 
should  not  exceed  J^2  of  the  pole  pitch  referred  to  the 
commutator  surface.  Having  determined  the  width  of  the 
brush,  and  decided  upon  a  suitable  current  density,  the  total 
axial  length  per  brush  set  may  be  calculated,  and  the  length  of 
the  commutator  decided  upon. 

The  individual  brush  rarely  exceeds  2  in.,  measured  parallel  to 
the  axis  of  rotation,  and  when  a  greater  length  of  contact  surface 
is  required,  several  brush  holders  are  provided  on  the  one  spindle 
or  brush  arm.  Even  in  small  machines,  the  number  of  brushes 
per  set  should  not  be  less  than  two,  so  as  to  allow  of  examination 
and  adjustment  while  running.  The  final  check  in  the  matter 
of  commutator  design  is  the  probable  temperature  rise;  but  this 
will  be  again  referred  to  when  considering  losses  and  efficiency. 


COMMUTATION 


179 


The  curvv  Figs.  68  and  69,  give  respectively  the  contact  re- 
sistance and  tu  ^  of  potential  for  different  current  densities. 
Two  curves  are  pic  -d  in  each  case:  the  one  referring  to  a  hard 


20  30  40  50  CO 

Current  Density-  Amperes  per  Square  Inch 

FIG.  68. — Contact  resistance,  carbon  brushes. 


10 


.  Carbon 


20  30  40  50  60 

Current  Density    Amperes  per  Square  Inch 


70 


SO 


FIG.  69. — Pressure  drop  at  contact  surfaces,  carbon  brushes. 


quality  of  carbon,  and  the  other  to  the  much  softer  electro- 
graphitic  carbon  which  has  a  lower  resistance  and  may  be  worked 
at  a  higher  current  density.  An  average  pressure  of  1.5  Ib. 


180          PRINCIPLES  OF  ELECTRICAL  DESIGN 

per  square  inch  between  contact  surfaces  has  bf  2  assumed  in 
order  to  avoid  the  plotting  of  a  large  numb'-  ui  curves.  For 
high-voltage  machines,  the  harder  qualit-"  JL« '  carbon  will  gener- 
ally be  found  most  suitable.  For  low ._  _ jltages,  economy  may 
frequently  be  effected  by  using  the  graphitic  brushes  with  cur- 
rent densities  as  high  as  60  amp.  to  the  square  inch  of  contact 
surface.  It  is  an  interesting,  but  not  very  clearly  explained, 
fact  that  the  temperature  ris'j  of  the  negative  brushes  is  greater 
than  that  of  the  positive  brushes.  In  other  words,  the  watts 
lost  are  greater  when  the  current  flow — according  to  the  popular 
conception — is  from  carbon  to  copper,  than  when  it  is  from 
copper  to  carbon.  Tne  resistances  given  in  Fig.  68  have  been 
averaged  for  the  +  and  —  brushes. 

On  low-voltagj  dynamos,  when  the  current  to  be  collected  is 
very  large,  copper  brushes  must  be  used.  The  resistance  of 
the  contact  between  brush  and  commutator  is  then  much  lower 
than  with  carbon  brushes  and  the  current  density  may  be  as 
high  as  200  amp.  per  square  inch  of  contact  surface.  The  con- 
tact-surface resistance  may  be  anything  between  0.0007  and 
0.0028  ohm  per  square  inch,  a  safe  figure  for  the  purpose  of 
calculating  the  brush  losses  being  0.002  ohm.  If  the  current 
density  at  the  contact  surface  is  150  amp.  per  square  inch  (a 
very  common  value),  the  total  loss  of  pressure  at  the  brushes 
will  be  0.004  X  150  =  0.6  volts,  instead  of  about  2  volts,  which 
is  usual  with  carbon  brushes. 

The  degree  of  hardness  of  the  copper  used  for  the  commutator 
segments  is  a  matter  of  importance;  an  occasional  soft  bar  will 
invariably  lead  to  sparking  troubles  because  of  unequal  wear. 
A  perfectly  true  cylindrical  commutator  surface  is  essential 
to  sparkless  running.  The  different  sets  of  brushes  should  be 
"st'aggered"  in  order  to  cover  the  whole  surface  of  the  commu- 
tator and  so  prevent  the  formation  of  grooves.  For  the  same 
reason,  and  also  to  ensure  more  even  wear  of  the  journals  and 
bearings,  some  end  play  should  be  allowed  to  the  shaft.  In 
large  machines  it  is  not  uncommon  to  provide  some  device,  in 
the  form  of  an  electromagnet  with  automatically  controlled 
exciting  coil,  to  ensure  that  the  desirable  longitudinal  motion 
of  the  rotating  parts  shall  be  obtained 

Owing  to  the  hardness  of  the  mica  insulation  relatively  to  that 
of  the  copper  bars,  there  is  a  tendency  for  the  mica  to  project 
slightly  above  the  surface  of  the  copper.  This  naturally  leads 


COMMUTATION  181 

to  sparking  troubles,  and  it  is  not  uncommon  to  groove  the 
commutator  between  bars,  cutting  down  the  mica  about  ^  in. 
below  the  surface,  leaving  an  air  space  as  the  insulation  between 
the  bars.  This  undercutting  process  may  have  to  be  repeated 
as  the  commutator  wears  down  in  use. 

As  the  effects  of  any  irregularities  on  the  commutator  suiface 
are  accentuated  by  high  speeds,  it  is  usual  to  limit  the  surface 
velocity  of  the  commutator  to  about  3,000  ft.  per  minute  when 
possible. .  The  diameter  of  the  commutator  in  large  machines 
is  generally  about  60  per  cent,  of  the  armature  diameter,  while, 
in  small  machines,  this  ratio  may  be  as  high  as  0.75. 

The  design  of  brushes  and  holders  is  a  matter  of  great  im- 
portance; as  a  general  rule,  it  may  be  said  that  the  lighter  the 
moving  parts  of  brush  and  holder,  the  better  the  conditions  in 
regard  to  sparking  when  the  surface  of  the  commutator  is  not 
absolutely  true. 

54.  Heating  of  Commutator — Temperature  Rise. — In  some 
cases  it  is  necessary  to  provide  special  means  of  ventilation  to 
keep  the  temperature  of  the  commutator  within  reasonable 
limits;  but  as  a  rule  a  sufficiently  large  cooling  surface  may  be 
obtained  without  unduly  increasing  the  size  and  cost  of  the 
commutator. 

The  losses  to  be  dissipated  consist  of: 

1.  The  PR  loss  at  brush-contact  surface. 

2.  The  loss  due  to  friction  of  the  brushes  on  commutator 
surface. 

The  PR  loss  in  the  commutator  segments  is  relatively  small 
and  can  usually  be  neglected. 

The  watts  lost  under  item  (1)  are  approximately  2/,  where  I 
is  the  total  current  taken  from  the  machine.  This  assumes  an 
average  value  of  2  volts  for  the  total  potential  drop  between 
commutator  and  carbon  brushes.  For  a  more  exact  determina- 
tion of  this  electrical  loss,  the  curves  of  Fig.  69  can  be  used.  As 
the  current  passing  into  all  the  positive  brushes  includes  the 
shunt  exciting  current,  an  allowance  should  be  made  for  this. 
Moreover,  the  assumed  condition  of  uniform  current  density 
over  brush-contact  surface  will  not  be  fulfilled  in  practice.  The 
uneven  distribution  of  current  density  will  increase  the  losses, 
and  it  will  be  advisable  to  add  about  25  per  cent,  to  the  values 
of  voltage  drop  as  read  off  the  curves  of  Fig.  69. 

The  losses  under  item  (2)  are  less  easily  calculated  because 


182          PRINCIPLES  OF  ELECTRICAL  DESIGN 

the  coefficient  of  friction  will  depend  not  only  upon  the  quality 
of  the  carbon  brush  but  also  on  the  condition  of  the  commutator 
surface. 

Let  P  =  the  pressure  of  the  brush  on  the  commutator,  in 
pounds  per  square  inch  of  contact  surface  (usually 
from  1  to  1%  Ib.)  ; 
c  =  the  coefficient  of  friction; 
A  =  the   total    area   of   brush  contact  surface    (square 

inches)  ; 

vc  =  the  peripheral  velocity  of  the  commutator  in  feet 
per  minute; 

then  the  friction  loss  is  cPAvc  foot-pounds  per  minute. 

If  DC  is  the  diameter  of  the  commutator  in  inches,  and  N  is 
the  number  of  revolutions  per  minute, 


The  friction  loss,  expressed  in  watts,  is 

cPAND*  X  746 
12  X  33,000 

The  value  of  c  for  a  good  quality  of  carbon  brush  of  medium 
hardness  may  lie  between  0.2  and  0.3;  but  this  coefficient  is 
not  reliable  as  it  depends  upon  many  factors  which  cannot 
easily  be  accounted  for. 

The  watts  that  can  be  dissipated  per  square  inch  of  commu- 
tator surface  will  depend  on  many  factors  which  cannot  be 
embodied  in  a  formula.  The  peripheral  velocity  of  the  com- 
mutator surface  will  undoubtedly  have  an  effect  upon  the  cool- 
ing coefficient;  but  the  influence  of  high  speeds  on  the  cooling 
of  revolving  cylindrical  surfaces  is  not  so  great  as  might  be 
expected.  The  design  of  the  risers  —  i.e.,  the  copper  connections 
between  the  commutator  bars  and  the  armature  windings  — 
has  much  to  do  with  the  effective  cooling  of  small  commutators; 
but  this  factor  is  of  less  importance  when  the  axial  length  of  the 
commutator  is  considerable. 

Some  designers  consider  only  the  outside  cylindrical  surface 
of  the  commutator  when  calculating  temperature  rise;  but  this 
leads  to  unsatisfactory  results  in  the  case  of  short  commutators. 
In  the  formula  here  proposed,  it  is  assumed  that  the  risers  add 
to  the  effective  cooling  surface  up  to  a  limiting  radial  distance 


COMMUTATION 


183 


of  2  in.;  that  is  to  say,  if  the  risers  are  longer  than  2  in.,  the  area 
beyond  this  distance  will  be  considered  ineffective  in  the  matter 
of  dissipating  heat  losses  occurring  at  the  commutator  surface. 
The  external  surface  of  the  carbon-brush  holders  is  helpful  in 
keeping  down  the  temperature  and  it  will  be  taken  into  account 
by  assuming  that  the  cooling  surface  of  the  commutator  is  in- 
creased by  an  amount  equal  to  2lcb  sq.  in.;  where  le  is  the  total 
axial  length  of  one  set  of  brushes,  and  b  is  the  total  number  of 
brush  sets. 


FIG.  70. — Cooling  surface  of  commutator. 

The  cooling  area,  as  indicated  in  Fig.  70,  will  therefore  con- 
sist of  the  cylindrical  surface  TrDcLe',  the  surface  of  the  risers 

£  (Dr2   -  Dc2);  the  surface  of  the  exposed  ends  (if  any)  of  the 

copper  bars,   of  value  --   (Dc2  —  D«2);   and   the   allowance    of 

2lcb  for  the  brush  holders. 

The  empirical  formula  here  proposed  for  calculating  the  tem- 
perature rise  of  the  commutator  is 

At  V 

(90) 


where  W  =  the  total  watts  to  be  dissipated. 

A  =  the  cooling  area  computed  as  above  (square  inches). 
vc  =  the  peripheral  velocity  of  the  cylindrical  surface  of 

the  commutator  in  feet  per  minute. 
T  =  the  temperature  rise  in  degrees  Centigrade. 


184          PRINCIPLES  OF  ELECTRICAL  DESIGN 

The  allowable  temperature  rise,  i.e.,  the  limiting  value  of  T 
in  formula  (90)  is  45°  to  50°C. 

The  mechanical  construction  of  the  commutator  is  a  matter 
of  great  importance,  and  when  the  peripheral  velocity  exceeds 
4,000  ft.  per  minute,  special  means  may  have  to  be  adopted 
to  ensure  satisfactory  operation.  For  instance,  if  the  axial 
length  is  great,  it  may  be  necessary  to  provide  one  or  more 
steel  rings  which  can  be  slipped  over  the  surface  of  the  bars  and 
shrunk  on,  to  prevent  displacement  of  the  bars  or  loosening  of 
the  mica  insulation  owing  to  vibration  or  centrifugal  action. 
These  mechanical  details  must,  however,  be  studied  elsewhere 
as  their  discussion  is  not  included  in  the  scope  of  this  book.  A 
sufficient  radial  depth  of  commutator  bar  must  be  provided  in 
order  that  the  strength  and  stiffness  may  be  sufficient  to  resist 
the  effects  of  centrifugal  force.  This  dimension  of  the  copper 
segments  should  include  an  allowance  of  %  to  %  in.  for  wear, 
as  the  commutator  must  be  large  enough  in  diameter  to  allow 
of  its  being  turned  down  occasionally  without  reducing  the 
cross-section  of  the  segments  to  a  dangerous  extent.  The 
mechanical  considerations  are  the  controlling  factors  in  this 
connection,  as  the  cross-section  is  usually  ample  to  carry  the 
required  current.  The  pitch  of  the  bars  at  the  commutator 
surface  should  not  be  less  than  0.2  in.,  because,  with  the  mica 
of  the  usual  thickness  (0.03  to  0.035  in.)  the  bar  would  be 
mechanically  unsatisfactory  if  the  thickness  were  reduced  below 
this  limit. 


CHAPTER  IX 

THE   MAGNETIC   CIRCUIT— DESIGN    OF   FIELD 
MAGNETS— EFFICIENCY 

55.  The  Magnetic  Circuit  of  the  Dynamo. — Once  the  total 
flux  per  pole  necessary  to  develop  the  required  voltage  is  known, 
it  is  an  easy  matter  to  design  the  complete  magnetic  circuit  and 
provide  it  with  a  suitable  winding  in  order  that  the  required 
flux  shall  enter  the  armature.  -The  method  of  procedure  is 
similar  to  that  adopted  in  the  design  of  a  horseshoe  lifting  magnet 
(see  Art.  16,  Chap.  Ill),  and  for  this  reason  the  subject  will  be 
dealt  with  very  briefly  in  this  chapter. 


SECTION  THROUGH 

A-B 


FIG.  71. — Magnetic  circuit  of  multipolar  dynamo. 

Fig.  71  shows  the  flux  paths  in  a  multipolar  dynamo.  If  we 
know  the  flux  density  in  the  iron  at  all  parts  of  the  magnetic 
circuit  and  the  average  lengths,  y,  c,  and  a,  of  the  flux  paths  per 
pole  in  the  yoke,  pole  cores,  and  armature,  respectively,  we  can, 
by  referring  to  the  B-H  curves  of  the  materials  used  in  the  con- 
struction, easily  calculate  the  ampere-turns  necessary  to  over- 
come the  reluctance  of  these  portions  of  the  magnetic  circuit. 
The  greatest  part  of  the  total  reluctance  is  in  the  air  gap  and 

185 


186          PRINCIPLES  OF  ELECTRICAL  DESIGN 

teeth;  but  the  amount  of  excitation  required  to  send  the  flux 
through  air  gap,  teeth,  and  slots,  has  already  been  calculated, 
and  may  be  read  off  the.  curve  a  of  Fig.  49,  page  133.  The  air- 
gap  density  to  be  used  in  obtaining  this  component  of  the  total 
field  ampere-turns  will  be  the  maximum  value  of  the  air-gap 
density  as  obtained  from  the  flux  curve  A  of  Fig.  51  (for  open- 
circuit  conditions). 

The  necessary  cross-section  of  iron  in  the  various  parts  of  the 
magnetic  circuit  is  readily  calculated  if  the  leakage  factor  can 
be  estimated;  but  the  length  of  tne  pole  core  (the  dimension  c 
in  Fig.  71)  will  depend  upon  the  number  of  ampere-turns  re- 
quired, and  therefore  on  the  length  of  the  air  gap,  which  must  be 
decided  upon  at  an  early  stage  in  the  design  (see  end  of  Art.  36, 
page  119). 

Let  (SI)gt  be  the  ampere-turns  required  at  full  load  for  the 
air  gap,  teeth,  and  slots;  then,  if  we  assume  a  depth  of  winding 
on  the  field  coils  of  1%  in.,  a  winding  space  factor  of  0.5,  and  a 
current  density  'of  1,000  amp.  per  square  inch  of  copper  cross- 
section,  the  length  of  the  winding  space  (which  is  approximately 
equal  to  the  length  of  c  of  Fig.  71)  would  be: 

(SI)at 


'   875 

If,  now,  we  make  the  further  assumptions  that  (SI)gt  is  50 
per  cent,  greater  than  the  ampere-turns  necessary  to  overcome 
the  reluctance  of  the  actual  air  clearance  of  length  d  in.,  and 
that  the  air-gap  density  B0  =  8,000  gausses,  we  may  write: 

2^fl^  =  «  X  2.54  X  8,000 

J.  .O 

and,  putting  875c  in  place  of  (SI)gt,  we  get  the  relation 

2.54  X  8,000  X  1.5 

0.4*-  X  875 
=  285  (approximately)  (92) 

For  a  preliminary  calculation  of  the  ampere-turns  required 
for  the  complete  magnetic  circuit,  a  value  of  c  (the  length  of  the 
pole)  rather  greater  than  as  calculated  by  formula  (91)  or  (92) 
may  be  selected.  This  dimension  will  be  subject  to  modifica- 
tion if  it  is  afterward  found  that  the  cooling  surface  of  the  field 
windings  is  insufficient  to  prevent  an  excessive  rise  of  temperature. 

56.  Leakage  Factor  in  Multipolar  Dynamos.  —  Apart  from  the 
useful  flux  entering  the  armature  core,  there  will,  be  in  every 


THE  MAGNETIC  CIRCUIT 


187 


design  of  dynamo,  some  leakage  flux  between  pole  shoes  and 
between  pole  cores  which,  in  a  plane  normal  to  the  axis  of 
rotation,  will  follow  paths  somewhat  as  indicated  in  Fig.  72. 
The  amount  of  this  leakage  flux  is  not  easily  calculated;  but  it 
can  be  approximately  predetermined  by  applying  the  conven- 
tional formulas  of  Art.  5,  Chap.  II,  or  by  the  graphical  method 
as  outlined  in  Art.  39,  Chap.  VII.  Other  approximate  graphical 
methods  are  used  by  designers1  and,  in  the  case  of  radical  de- 
partures from  standard  types,  some  such  method  of  estimating 
the  leakage  flux  must  be  adopted.  It  will,  however,  be  found 


FIG.  72. — Leakage  flux  in  multipolar  dynamo. 

that  the  leakage  factor  does  not  vary  appreciably  in  modern 
designs  of  multipolar  dynamos,  and  the  following  values  may  be 
adopted  for  the  purpose  of  determining  the  necessary  cross- 
sections  of  the  pole  cores  and  frame. 


Kw.  output  of  dynamo 

20  to  50 

50  to  150 

150  to  250 

250  to  400 

500  and  larger 


Leakage  coefficient 

1.15  to  1.3 
1.14  to  1.26 
1 . 13  to  1 . 23 
1.11  to  1.19 
1 . 10  to  1 . 16 


57.  Calculation  of  Total  Ampere-turns  Required  on  Field 
Magnets. — The  maximum  values  of  the  m.m.f.  curves  obtained 
by  the  method  followed  in  Chap.  VII  (see  Arts.  40  and  43) 
give  the  ampere-turns  per  pole  required  to  overcome  the  re- 
luctance of  the  air  gap,  teeth  and  slots.  The  balance  of  the 
total  ampere-turns  is  easily  calculated  since  the  lengths  and 
cross-sections  of  the  various  parts  of  the  magnetic  circuit  are 

1  See  p.  326  of  WALKER'S  "Specification  and  Design  of  Dynamo- 
electric  Machinery." 


188          PRINCIPLES  OF  ELECTRICAL  DESIGN 

known,  and  a  suitable  leakage  factor  may  be  selected  from  the 
table  in  the  preceding  article.  The  method  of  procedure  is 
exactly  as  explained  in  connection  with  the  design  of  a  horseshoe 
lifting  magnet  (see  Art.  16,  page  52),  and  it  will  be  again  fol- 
lowed in  detail  when  working  out  a  numerical  example  of  con- 
tinuous-current generator  design.  The  flux  path  of  average 
length  is  indicated  in  Fig.  71.  There  may  be  some  doubt  as  to 
what  is  the  proper  value  to  take  for  the  length  of  the  path  a  in 
the  armature  core  below  the  teeth,  because  the  flux  density  will 
be  less  uniform  in  this  part  of  the  magnetic  circuit  than  in  the 
poles  and  yoke.  As  a  matter  of  fact,  the  ampere-turns  neces- 
sary to  overcome  the  reluctance  of  the  armature  core  (apart 
from  the  teeth)  are  but  a  small  percentage  of  the  total,  because 
the  flux  density  must  necessarily  be  low  to  avoid  large  losses 
due  to  the  reversals  of  magnetic  flux.  It  is,  therefore,  something 
of  a  refinement  to  take  account  of  the  unequal  distribution  of 
the  flux  in  the  armature  core;  but  if  the  length  of  the  path  a 
of  Fig.  71  be  taken  as  one-third  of  the  pole  pitch  r,  a  more 
accurate  value  for  the  ampere-turns  will  be  obtained  than  if 
the  length  were  measured  along  the  curved  path  shown  in  the 
illustration.  It  is,  of  course,  understood  that  the  density  to  be 
used  in  the  calculation  is  the  maximum  flux  density  at  the 
section  midway  between  the  poles,  on  the  assumption  that  the 
flux  is  uniformly  distributed  over  this  section.  Thus,  if  $ 
is  the  useful  flux  per  pole  entering  the  armature  core,  and  Rd 

is  the  radial  depth  of  stampings  below  the  teeth,  the  maximum 

$ 
density  in  armature  core  to  be  used  in  the  calculations  is  op  7 

ZK  din 

where  ln  stands,  as  before,  for  the  net  axial  length  of  the 
armature. 

Solid-pole  shoes  are  rarely  used  in  connection  with  armatures 
having  open  slots.  With  semi-closed  slots,  or  even  with  open 
slots  if  the  air  gap  is  large,  the  eddy-current  losses  in  solid-pole 
shoes  may  be  very  small,  but  laminated  pole  pieces  are  now  the 
rule  rather  than  the  exception.  The  thickness  of  the  steel 
sheets  used  to  build  up  the  pole  pieces  is  usually  greater  than 
that  of  the  armature  punchings,  a  thickness  of  0.025  in.  being 
fairly  common.  In  small  machines  it  is  sometimes  economical 
to  construct  the  complete  pole  of  sheet-steel  stampings,  as  this 
dispenses  with  the  labor  cost  of  fitting  a  separate  built-up  pole 
piece  on  the  solid  pole  core. 


THE  MAGNETIC  CIRCUIT 


189 


Referring  to  Fig.  49  on  page  133,  the  curve  (a)  was  plotted 
by  assuming  different  values  of  air-gap  density;  it  shows  the 
relation  between  the  ampere-turns  required  for  air  gap,  teeth 
and  slots,  and  the  air-gap  density  over  the  slot  pitch  at  the 
center  of  the  pole  face.  We  are  now  in  a  position  to  plot  an 
open-circuit  saturation  curve  which  shall  include  the  ampere- 
turns  necessary  to  overcome  the  reluctance  of  all  parts  of  the 
magnetic  circuit;  but  instead  of  giving  the  relation  between  total 
ampere-turns  per  pole  and  the  air-gap  density,  it  will  be  more 
convenient  to  plot  a  curve  connecting  field  ampere-turns  and 
e.m.f.  generated  in  the  armature.  This  can  easily  be  done  since 
we  know  the  total  flux  per  pole  and  the  generated  e.m.f.  corre- 
sponding to  the  value  Ba  of  the  maximum  air-gap  density  (Fig. 
49).  Thus,  in  Fig.  73  the  distance  OPa  is  the  same  as  in  Fig. 
49,  but  the  vertical  scale  has  been  altered  so  that  the  correspond- 
ing value  Eo  as  read  off  the  dotted  curve  of  Fig.  73  now  stand  for 
the  volts  developed  in  the  armature  by  the  cutting  of  the  flux, 
instead  of  the  air-gap  density  under  the  center  of  the  pole  face. 
The  full-line  curve  of  Fig.  73  is  the  open-circuit  characteristic 
of  the  whole  machine.  It  gives 
the  connection  between  field  am- 
pere-turns per  pole  and  the  termi- 
nal voltage  on  open  circuit,  on  the 
understanding  that  the  speed  is 
constant.  The  additional  ampere- 
turns  required  to  overcome  the 
reluctance  of  the  field  poles,  yoke, 
and  armature  core,  account  for  the  p 
space  between  the  full-line  and  c 
dotted  curves.  This  no-load  sat- 
uration curve  for  the  complete 
machine  has  been  re-drawn  in  Fig.  74. 


Ampere  Turns  per  Pole 
FIG.  73. 


Here  OE0  is  the  termi- 
nal voltage  on  open  circuit;  OEt  is  the  terminal  voltage  at  full 
load  (the  machine  is  assumed  to  be  over-compounded) ;  and  OEd 
is  the  necessary  developed  voltage  at  full  load,  i.e.,  the  voltage 
that  must  be  generated  in  the  armature  conductors  by  the 
cutting  of  the  flux  in  order  that  the  terminal  voltage  at  full 
load  shall  be  OEt. 

Draw  a  straight  line  connecting  the  origin,  0,  of  the  curve  and 
the  point  F  corresponding  to  the  no-load  voltage,  and  produce 
this  to  G  where  it  meets  the  horizontal  line  representing  full- 


190 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


load  terminal  voltage.  Then,  since  the  ampere-turns  on  the 
shunt  at  no  load  are  OA,  they  will  obviously  have  increased  to 
OB  at  full  load  on  account  of  the  higher  terminal  voltage  (the 
"long  shunt"  connection  is  here  assumed).  The  ampere-turns 
necessary  to  produce  the  required  full-load  flux  will  be  OC; 
but  the  field  excitation  must  be  greater  than  this  in  order  to 
balance  the  distortional  and  demagnetizing  effects  of  the  arma- 
ture current.  It  was  found  that  the  ampere-turns  necessary 
to  counteract  the  effects  of  the  armature  current  were  repre- 
sented by  the  distance  PbPa  in  Fig.  49  (Art.  42,  page  133) .  These 


Ed 


Full  Load  Developed  Volts 

Full  Load  Term'l  Volts  G/ 


No  Load  Volts 


Shunt  (Full  Load)  <      Series   _^ 


D 


O  ABC 

Ampere  Turns  per  Pole 

FIG.  74. — Open-circuit  saturation  curve  of  dynamo. 


SI  to  Compensate 
for  Demagnetising 
aaid  Distortional 
Effect  of 
Armature 


ampere-turns  had  to  be  put  on  the  field  poles,  not  to  increase 
the  air-gap  flux  and  thus  develop  a  higher  voltage,  but  merely 
to  counteract  the  effects  of  the  armature  current  and  restore 
the  air-gap  flux  to  its  original  value  on  open  circuit.  It  is 
therefore  correct  to  say  that  additional  ampere-turns  approxi- 
mately equal  to  this  amount  must  be  added  to  the  field  wind- 
ings in  order  that  the  necessary  flux  shall  be  cut  by  the  armature 
conductors.  This  addition  is  shown  in  Fig.  74,  where  the 
distance  CD  has  the  same  value  as  PbPa  in  Fig.  49.  It  follows 
that  OD  represents  the  total  ampere-turns  required  per  pole 
at  full  load.  Of  this  total  amount,  OB  will  be  due  to  the  shunt 
winding,  and  the  balance,  BD,  must  be  provided  by  the  series 
winding. 

58.  Arrangement  and  Calculation  of  Field  Windings. — Since 
the  ampere-turns  required  in  the  shunt  winding  have  now  been 


THE  MAGNETIC  CIRCUIT  191 

determined,  the  calculation  of  the  size  of  wire  for  a  given  voltage 
may  be  proceeded  with  exactly  as  explained  in  connection  with 
the  winding,  of  magnet  coils  (see  Art.  10,  Chap.  II).  The  allow- 
able cooling  surface  is  not  quite  the  same  as  for  the  coils  of  lifting 
magnets,  because  the  fanning  effect  of  the  rotating  armature  is 
to  some  extent  beneficial;  but  it  will  be  convenient  to  consider 
the  heating  effects  of  the  shunt  and  series  coils  together,  and 
the  question  of  cooling  coefficients  for  use  in  predetermining 
temperature  rise  will  therefore  be  taken  up  later. 

Shunt  Field  Rheostat. — Even  when  the  machine  is  compounded 
by  the  addition  of  a  series  winding,  it  is  usual  to  provide  an 
adjustable  resistance  in  series  with  the  shunt  winding.  This 
field  rheostat  allows  of  the  excitation  being  kept  constant  not- 
withstanding the  fact  that  the  shunt  winding  will  not  have  the 
the  same  resistance  when  cold  as  it  will  have  when  a  continuous 
run  of  several  hours'  duration  has  raised  the  temperature  of 
the  coils.  The  rheostat  also  allows  of  final  adjustments  being 
made  after  the  machine  has  been  built  and  tested. 

In  compound-wound  generators  it  is  customary  to  allow  a 
voltage  drop  in  the  rheostat  amounting  to  15  or  20  per  cent, 
of  the  total  terminal  pressure;  and  a  sufficient  number  of  con- 
tacts should  be  provided  to  avoid  a  variation  of  more  than  % 
to  1  per  cent,  change  of  voltage  when  cutting  in  or  out  sections 
of  the  rheostat. 

The  size  of  wire  for  the  shunt  field  coils  should  therefore  be 
calculated  on  the  assumption  that  the  impressed  voltage  is  15 
to  20  per  cent,  less  than  the  terminal  voltage  of  the  machine. 
It  will  generally  be  found  desirable  to  connect  the  windings  on 
all  the  poles  in  series.  With  shunt-wound  dynamos,  the  field 
rheostat  plays  a  more  important  part:  it  must  be  designed  to 
give  the  required  variation  in  field  strength  between  no  load  and 
full  load,  at  constant  speed,  or,  in  the  case  of  a  motor,  to  provide 
for  the  required  speed  variation.  The  amount  by  which  the 
excitation  has  to  be  varied — apart  from  the  requirements  to 
compensate  for  the  effects  of  temperature  changes — may  be 
determined  by  reference  to  the  saturation  curve  as  drawn  in 
Fig.  74. 

In  a  well-designed  machine,  the  PR  losses  in  the  shunt  field 
winding  should  not  greatly  exceed  the  values  given  below,  where 
the  loss  is  expressed  as  a  percentage  of  the  rated  output  of  the 
dynamo : 


192          PRINCIPLES  OF  ELECTRICAL  DESIGN 

Output  of  machine,  kilowatts       Exciting  current,  percentage  of 

of  total  current 

10  3.5 

20  3.0 

50  2.4 

100  2.0 

200  1.7 

300  1.6 

500  1.5 

1,000  and  larger  1 . 3  to  1 . 0 

Series  Windings. — The  series  winding  on  the  field  magnets 
carries  the  main  current  from  the  machine  and  thus  adds  to  the 
constant  excitation  of  the  shunt  coils  a  number  of  ampere-turns 
generally  in  accordance  with  the  demand  for  a  stronger  field. 
It  is  not  usual  to  wind  the  series  turns  on  the  outside  of  the 
shunt  wire;  but  this  may  be  done  in  small  machines.  The  series 
turns  are  usually  placed  at  one  end  of  the  pole,  either  up  against 
the  pole  shoe,  or — more  commonly — near  the  yoke  ring.  Space 
must,  therefore,  be  left  for  the  series  winding  at  the  time  when 
the  dimensions  of  the  shunt  coil  are  decided  upon.  The  total 
winding  space  available  may  be  divided  in  proportion  to  the 
ampere-turns  required  on  the  shunt  and  series  coils  respectively. 
The  coils  near  the  yoke  ring,  in  a  machine  with  revolving  arma- 
ture, are  frequently  made  to  project  from  the  pole  core  farther 
than  the  coils  near  the  pole  shoe,  partly  because  the  space 
available  between  the  poles  increases  with  the  radial  distance 
from  the  center,  but  also  because  the  cooling  effect  of  the  air 
thrown  from  the  rotating  armature  will  be  greater  if  the  field 
windings  are  stepped  out  in  this  manner. 

The  construction  and  insulation  of  field  windings  deserves 
careful  attention;  but  for  details  of  this  nature,  the  designer 
must  rely  largely  upon  the  practice  of  manufacturing  firms  and 
his  own  common  sense.  The  pressures  to  be  considered  in 
D.C.  designs  are  not  high,  and  the  chief  points  requiring  atten- 
tion are  the  proper  arrangement  and  the  insulation  of  the  start- 
ing and  finishing  ends  of  the  coils.1 

The  size  of  conductors  for  use  in  the  series  winding  may  be 
determined  by  considerations  of  permissible  voltage  drop,  or, 
if  this  is  unimportant,  the  temperature  rise  will  be  the  determin- 

1  Much  useful  information  regarding  the  insulation  of  windings  will  be 
found  in  Chap.  IV  of  "Insulation  and  Design  of  Electrical  Windings,"  by 
A.  P.  M.  FLEMING  and  R.  JOHNSON  (LONGMANS  &  Co.). 


THE  MAGNETIC  CIRCUIT 


193 


ing  factor.  In  the  latter  case,  the  allowable  current  density 
will  be  about  the  same  as  in  the  shunt  coils,  unless  the  series 
turns  are  next  to  the  pole  shoe,  in  which  case  a  slightly  higher 
density  would  be  permissible  because  of  the  better  ventilation. 
If  the  current  to  be  carried  exceeds  100  amp.,  the  coils  may  be 
made  of  flat  copper  strip  wound  edgewise  by  means  of  a  special 
machine.  For  smaller  currents,  cotton-covered  wires  of  square 
or  rectangular  section  are  commonly  used;  the  round  wire  being 
rarely  employed,  unless  the  diameter  is  less  than  that  of  No.  8 
B.  &  S.  gage. 

On  account  of  the  method  of  connecting  the  series  coils  be- 
tween adjacent  poles  on  a  multipolar  dynamo,  there  will  be  an 
odd  number  of  turns  per  pair  of  poles,  or  a  whole  number  plus 
a  half  turn  on  each  pole.  This  will  easily  be  understood  by  re- 
ferring to  Fig.  75  which  is  supposed  to  show  a  portion  of  the 


FIG.  75. — Diagram  of  series  field  winding. 

crown  of  poles,  looking  down  through  the  yoke  ring  onto  the 
cylindrical  surface  of  the  armature.  The  number  of  ampere- 
turns  required  per  pole  being  known,  the  number  of  turns  in 
the  series  winding  can  easily  be  calculated.  It  may  not  be 
possible  to  put  this  exact  number  on  the  pole,  and  a  slightly 
greater  number  of  turns  is,  therefore,  provided,  the  excess  of 
current  being  shunted  through  a  diverter.  This  is  merely  a  re- 
sistance connected  as  a  shunt  to  the  series  winding.  Thus  if  the 
required  series  ampere-turns  per  pole  are  2,000,  and  the  current 
300  amp.,  the  calculated  number  of  turns  per  pole  is  6.66. 
Since  6j-£  turns  will  not  be  sufficient,  the  winding  may  consist 
of  7H  turns,  and  the  current  required  through  the  coils  is,  there- 
fore, 2,000/7.5  =  267  amp.  The  balance  of  33  amp.  must  be 
shunted  through  the  diverter,  the  resistance  of  which  is  easily 
calculated  after  determining  the  resistance  of  the  series  coils 
from  the  known  cross-section  and  computed  length  of  the  winding. 
69.  Temperature  Rise  of  Field  Coils. — On  account  of  the 
proximity  of  the  shunt  and  series  windings,  it  is  advisable  to 

13 


194 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


consider  the  joint  losses  in  connection  with  the  entire  cooling 
surface.  The  reader  is  referred  to  Art.  11,  Chap.  II,  where  the 
heating  of  magnet  coils  was  discussed.  The  problem  of  keeping 
the  temperature,  rise  of  field  coils  within  safe  limits  (40°  to  45°C.) 
is  complicated  by  the  fact  that  the  fanning  action  of  the  rotating 
armature  will  have  an  effect  upon  the  cooling  coefficient;  but 
this  has  been  taken  into  account  in  the  curves  of  Fig.  76.  The 


0.011 


0.009 


«?  0.007 

1 

'8 


0.006 


l 


0.005 


cT 


1000  2000  3000  4000  5000 

Peripheral  Speed  of  Armature  -Feet  per  Minute 

FIG.  76. — Cooling  coefficients  for  field  coils  of  dynamos. 


curve  marked  A  applies  to  machines  with  wide  spacing  between 
poles,  and  good  ventilation,  while  the  curve  B  should  be  used 
when  the  main  poles  are  close  together,  or  when  commutating 
poles  interfere  with  the  free  circulation  of  air  round  the  main 
windings.  The  coefficient  obtained  from  the  curves,  being 
watts  per  square  inch  per  degree  rise,  is  the  reciprocal  of  the  coef- 
ficient k  used  in  the  chapter  on  magnet  design;  but  the  cooling 
surface  considered  is  the  same,  namely,  the  total  external  surface 
of  the  winding,  including  the  two  ends  and  also  the  inner  surface 


THE  MAGNETIC  CIRCUIT  195 

near  the  iron  pole  core.  The  cooling  coefficient  will  necessarily 
depend  upon  the  type  and  size  of  the  machine,  and  it  should,  if 
possible,  be  determined  from  tests  made  on  machines  generally 
similar  to  the  one  being  designed.  The  modern  tendency  in 
design  is  all  toward  increased  output  by  improvements  in  the 
qualities  of  materials  and  in  methods  of  ventilation.  Field 
coils  are  now  frequently  built  with  seetionalized  windings  so 
arranged  that  the  air  has  free  access,  not  only  between  the  sub- 
divisions of  the  winding,  but  also  between  the  inside  of  the  coils 
and  the  pole  core.  The  gain  is  not  always  proportionate  to  the 
total  cost  and  space  required;  but  the  cooling  coefficients  given 
in  Fig.  76  would  not  be  applicable  to  such  designs  without  modi- 
fication. Each  manufacturer  has  his  own  data  to  guide  him  in 
his  calculation  of  new  designs;  but  even  if  such  data  were 
available  for  publication,  it  would  be  of  little  value  without  the 
experience  which  enables  the  designer  to  apply  it  intelligently 
to  a  practical  case. 

60.  Efficiency. — The  efficiency  of  a  dynamo  is  the  ratio  of 
power  output  to  power  input,  or, 

_  output 

Efficiency  ==  output  +  losses 

In  computing  the  total  losses,  an  estimate  has  to  be  made 
of  the  power  lost  through  windage  and  bearing  friction.  It  is 
almost  impossible  to  predetermine  these  quantities  accurately. 
The  loss  due  to  air  friction  will  depend  upon  the  design  of  the 
armature  and  arrangement  of  poles  and  frame,  apart  from  the 
actual  surface  velocity;  while  the  bearing  friction  will  depend 
upon  the  number  and  size  of  the  bearings,  the  method  of  lubri- 
cation, the  weight  of  the  rotating  parts,  and  the  method  of 
coupling  to  the  prime  mover.  The  factors  to  be  taken  into 
account  are  so  numerous  and  so  difficult  to  determine  that,  in 
the  case  of  new  designs  or  departures  from  standard  types,  it  is 
usual  to  group  these  losses  together  and  make  a  reasonable 
allowance  for  them  in  the  calculations  of  efficiency.  The  fric- 
tion losses  will  increase  with  the  surface  velocities;  but  since 
the  volume,  and  therefore  the  weight,  of  the  rotating  armature 
of  a  machine  of  given  output  will  decrease  with  increase  of  speed, 
it  is  found  that  the  total  friction  losses  may  be  expressed  as  a 
percentage  of  the  total  output,  and  this  percentage  will  not 
vary  greatly  in  machines  of  different  outputs  and  speeds.  The 


196          PRINCIPLES  OF  ELECTRICAL  DESIGN 

following  figures  indicate  approximately  the  losses  due  to  wind- 
age and  bearing  friction  in  modern  types  of  dynamos. 

Rated  kw.  output  Friction  losses  (per  cent.) 

10  3.0 

30  2.5 

60  2.0 

100  1.5 

200  1.0 

500  0.75 

Large  machines  0 . 6 

Closer  Estimate  of  Hysteresis  and  Eddy-current  Losses  in 
Armature  Teeth. — For  the  purpose  of  calculating  the  armature 
losses  with  sufficient  accuracy  to  determine  whether  or  not  the 
temperature  rise  is  likely  to  be  excessive,  it  was  suggested  in 
Art.  31  (page  103),  that  the  average  value  of  the  apparent  tooth 
density  be  used  in  calculating  the  iron  loss  in  the  teeth.  When 
the  tooth  density  is  very  high,  or  the  taper  of  the  tooth  consider- 
able, this  method  will  not  yield  very  accurate  results.  It  is  the 
actual  tooth  density  which,  together  with  the  frequency,  will 
determine  the  losses  per  pound  in  a  given  quality  of  steel  punch- 
ings;  and  this  actual  tooth  density  may  be  read  off  a  curve  plotted 
from  the  formulas  derived  in  Art.  37,  Chap.  VII. 

The  tooth  density  with  which  we  are  concerned  in  the  calcu- 
lations of  power  losses,  is  obviously  the  maximum  density,  and 
this  will  occur  when  the  tooth  is  in  the  zone  of  maximum  air 
gap  density,  the  value  of  which  can  be  read  off  the  full-load 
flux  distribution  curve,  C,  derived  as  explained  in  Art.  43, 
Chap.  VII. 

When  the  tooth  is  not  of  the  same  cross-section  throughout 
its  length,  the  question  arises  as  to  what  particular  value  of  the 
actual  tooth  density  should  be  taken  for  the  purpose  of  calcu- 
lating the  iron  losses.  The  tooth  might  be  divided  into  a 
number  of  imaginary  sections  concentric  with  the  shaft,  and  the 
watts  lost  in  the  elemental  sections  could  be  calculated  and 
totalled;  but  this  would  be  a  lengthy  and  tedious  process,  and 
the  following  approximation  will  usually  give  results  of  suffi- 
cient accuracy  for  practical  purposes. 

First  calculate  the  actual  flux  density  at  the  root  of  the  tooth 
(see  Art.  37,  page.  119),  and  then  again,  at  two  other  sections, 
namely,  near  the  top  where  the  circumference  of  the  "  equivalent" 


THE  MAGNETIC  CIRCUIT  197 

smooth  core  armature  would  cut  through  the  tooth,  and  also  at 
a  point  midway  between  these  two  extremes.  These  sections 
are  shown  in  Fig.  38,  page  122;  and  if  the  assumption  is  made  that 
no  flux  lines  either  enter  or  leave  the  sides  of  the  tooth,  the  densi- 
ties at  the  middle  and  top  of  the  tooth  can  readily  be  expressed 
in  terms  of  the  root  density  since  they  will  vary  inversely  as  the 
cross-section  of  the  tooth.  If,  now,  the  iron  loss  in  watts  per 
pound  is  read  off  the  curves  of  Fig.  34,  page  102,  for  the  three 
selected  values  of  the  tooth  density,  the  average  loss  per  pound 
multiplied  by  the  total  weight  of  iron  in  the  teeth,  will  represent, 
within  a  reasonable  degree  of  accuracy,  the  total  loss  in  the 
teeth  of  the  machine. 

A  high  tooth  density  is  an  advantage  from  the  point  of  view 
of  field  distortion.  It  can  easily  be  understood  that  a  high  flux 
density,  by  saturating  the  teeth,  will  have  a  tendency  to  resist 
the  changes  in  the  air-gap  flux  distribution,  brought  about  by 
the  cross-magnetizing  effect  of  the  armature  currents;  and  it  is 
therefore  advisable  to  check  the  approximate  estimate  of  tooth 
losses  with  the  results  obtained  by  the  more  exact  method  of 
calculation. 

In  predetermining  the  efficiency,  it  is  important  that  all  the 
power  losses  in  the  machine  be  taken  into  account.  These  must 
include  the  losses  in  rheostats  or  diverters,  which  may  be  con- 
sidered as  part  of  the  machine.  A  complete  list  of  the  losses  to 
be  evaluated  when  predetermining  the  efficiency  will  be  found 
in  the  following  chapter,  where  a  numerical  example  will  illus- 
trate in  detail  the  steps  to  be  followed  in  designing  a  dynamo. 

As  a  check  on  calculations,  the  efficiencies  in  the  following 
tables  may  be  referred  to.  They  represent  a  fair  average  of 
what  may  be  expected  in  a  machine  of  modern  design. 


FULL-LOAD  EFFICIENCIES  OF  DYNAMOS 

Kw.  output  Efficiency,  per  cent. 

10  86.0 

20  87.6 

30  88.8 

40  89.7 

50  90.3 

75  91.2 

100  91.6 

200  92.0 

500  and  larger  93  to  95 


198          PRINCIPLES  OF  ELECTRICAL  DESI  N 

EFFICIENCIES  OF  DYNAMOS  WHEN  NOT  DELIVERING  FULL-LOAD   CURRENT 


Output  as  fraction  of  full  load 

Size  of  machine  (Rated  kw.  output) 

10  kw. 

200  kw. 

y±  

70.0  per  cent. 
80.0    "      " 
84.5    "      " 
86.0    "      " 

86  per  cent. 
91    "       " 
92    " 
92    "       " 

y>. 

%  

Full  load  

CHAPTER  X 

PROCEDURE  IN  DESIGN  OF  D.C.  GENERATOR- 
NUMERICAL  EXAMPLE 

61.  Introductory. — Since  the  procedure  about  to  be  followed 
in  working  out  a  numerical  example  in  dynamo  design  is  not 
likely  to  meet  with  the  approval  of  every  practical  designer,  it 
is  well  to  remember  that  an  attempt  is  here  made  to  base  the 
work  on  fundamental  principles  and  show  how  these  principles 
may  be  applied  in  the  detailed  design  of  dynamo-electric 
machinery.  The  method  here  presented  is  one  that  will  yield 
very  satisfactory  results  when  developing  new  types  of  machines, 
or  when  no  account  need  be  taken  of  existing  patterns  or  tools. 
The  practical  designer  is  usually  compelled  to  use  stock  frames 
and  armature  punchings,  and  adapt  these  to  the  requirements 
of  the  specification.  He  must  effect  some  sort  of  compromise 
between  the  ideal  design  and  a  design  that  will  comply  with 
manufacturing  conditions.  In  the  method  here  followed,  the 
assumption  is  made  that  the  designer  is  given  a  free  hand  to 
produce  a  machine  that  shall,  in  all  respects,  be  suitable  for 
the  work  it  has  to  perform,  and  of  which  the  cost  and  efficiency 
shall  be  generally  in  accordance  with  present-day  requirements. 
The  various  steps  in  the  electrical  design  of  a  D.C.  generator  will 
be  followed  in  logical  sequence,  and  if  the  work  appears  unneces- 
sarily detailed  and  drawn  out,  it  must  be  remembered  that 
the  method  has  an  educational,  apart  from  a  practical,  value; 
it  illustrates  the  application  of  theoretical  principles  to  a  con- 
crete case,  and  shows  how  the  practice  of  engineering  is  largely 
a  matter  of  scientific  guesswork.  The  experienced  designer  will 
be  able  to  skip  many  of  the  intermediate  steps  here  purposely 
included;  because  he  will  be  able  to  rely  on  the  engineering  judg- 
ment he  has  acquired  during  years  of  practice  in  similar  work. 
The  point  that  must  never  be  lost  sight  of  is  that,  when  an  engi- 
neer makes  a  guess  in  respect  to  a  dimension  or  any  quantity 
of  doubtful  or  indeterminate  value,  he  is  always  able  to  check 

199 


200          PRINCIPLES  OF  ELECTRICAL  DESIGN 

the  accuracy  of  his  estimate  by  satisfying  himself  that  the 
results  obtained  accord  with  the  known  laws  of  physics. 

Since  the  design  of  commutating  poles  was  treated  at  some 
length  in  the  chapter  on  commutation  (see  Art.  51,  Chap.  VIII), 
it  is  proposed  to  select  for  the  purpose  of  illustration  a  machine 
of  comparatively  small  output,  and  endeavor  to  obtain  satis- 
factory commutation  without  the  addition  of  interpoles. 

62.  Design  Sheets  for  75-kw.  Multipolar  Dynamo. — The 
particulars  contained  in  the  following  design  sheets  are  more 
than  sufficient  for  the  needs  of  the  practical  designer;  but  they 
serve  a  useful  purpose  as  a  guide  in  making  the  calculations. 
The  items  are  numbered  for  easy  reference,  and  it  will  be  found 
convenient  to  calculate  the  required  dimensions  and  quantities 
generally  in  the  order  given,  although  the  particular  arrange- 
ment here  adopted  need  not  be  adhered  to  rigidly.  Two  columns 
are  provided  for  the  numerical  values,  and  they  are  supposed 
to  be  filled  in  as  the  work  proceeds.  The  first  of  these  columns 
is  to  be  used  for  assumed  values  or  preliminary  estimates;  while 
the  last  column  is  reserved  for  the  corrected  final  values. 

The  actual  calculations  will  follow  the  design  sheets,  and 
they  will  be  shown  in  sufficient  detail  to  be  self-explanatory. 
The  calculation  of  items  of  which  the  numerical  values  are  ob- 
viously derived  from  previously  obtained  quantities  will  not 
always  be  shown  in  detail.  The  design  sheets  should  be  fol- 
lowed item  by  item,  and  where  the  method  of  calculation  is 
not  clear,  the  succeeding  pages  may  be  consulted  for  explanations 
and  references  to  the  text. 

The  machine  to  be  designed  is  for  a  continuous  output  of 
75  kw.  The  terminal  voltage  on  open  circuit  is  220;  but  it  is 
required  to  raise  this  to  230  at  full  load  in  order  to  compensate 
for  loss  of  pressure  in  cables  between  the  dynamo  and  the  place 
where  the  power  is  utilized.  The  machine  is  therefore  over- 
compounded.  It  is  to  be  belt-driven  at  a  constant  speed  of 
600  revolutions  per  minute.  The  temperature  rise  is  not  to 
exceed  40°C.  after  a  full-load  run  of  not  less  than  6  hr.  duration. 

All  numerical  calculations  have  been  worked  out  on  the  slide 
rule,  and  scientific  accuracy  in  the  results  is  not  claimed. 


PROCEDURE  IN  DESIGN  OF  D.C.  GENERATOR     201 

DESIGN  SHEET  FOR  CONTINUOUS-CURRENT  GENERATOR 


SPECIFICATION 

Symbols 

Preliminary 
or  assumed 
values 

Final 
values 

1    Kw  output 

75 

2.   No-load  terminal  voltage 

220 

3    Full-load  terminal  voltage 

230 

4.  Speed,  r  p  m 

600 

5.  Permissible  temperature  rise 

T  = 

40°C 

PRELIMINARY  ASSUMPTIONS  AND  CALCULATIONS 

6     Number  of  poles  
7.    Frequency  
8.    Per  cent,  armature  surface  covered  by  poles  
9.  Specific  loading  of  armature  
10.   Type  of  armature  winding  (series  or  multiple)  
11.  Apparent  air-gap  density  (open  circuit)  
12.  Line  current  
13.  Armature  current  (per  circuit)  
14.  Armature  diameter  (inches)  
15.  Peripheral  velocity  (feet  per  minute)  
16.  Total  number  of  face  conductors  
17.  Armature  ampere-turns  per  pole  
18.  Length  of  air  gap.  ...        .    . 

P  = 
/  = 
r  = 
Q  = 

B8- 

7  = 

Ic  = 
D  = 
v  = 
Z  = 

5  = 

4 
20 
0.72 
475 

8,000 

83.4 
19.63 

350 
3,650 
0  228 

4 
20 
0.72 
461 
lap. 

326 
82.6 
19.5 
3,070 
342 
3,565 
0.25 

19.  No-load  flux  per  .pole  (maxwells)  
20.  Pole  pitch  (inches)  
21.  Pole  arc  (inches)  
22.  Area  under  pole  face  (square  inches)  
23.  Axial  length  of  armature  (gross) 

*  = 

T  =* 

la  = 

6,290,000 

11.05 
122 
11    1 

6,430,000 
15.34 
11 

11 

24.  Axial  length  of  pole  face  
25.  Cross-section  of  each  armature  conductor   (square 
inches)  

10.6 
0  0402 

10.5 
0.039 

26.  Dimensions  of  armature  conductor  (inches) 

!4X$i« 

27.  Number  of  slots  

57 

57 

28.  Number  of  inductors  per  slot 

6 

6 

29.  Slot  pitch  (inches)  

X  = 

1.076 

30.  Slot  width  (inches)  

8  = 

0.5 

31.  Slot  depth  (inches)  
32.  Tooth  width,  top  (inches)  
33.  Tooth  width,  average  (inches)  

d  = 

1.0 
0.576 
0.521 

34.  Tooth  width,  bottom  (inches)  
35.  Number  of  radial  cooling  ducts  
36.  Width  of  duct  (inches)  
37.  Net  length  of  armature  (inches)  
38.  Net  tooth  section  under  pole  (average)  
39.  Apparent  flux  density  in  teeth  (open  circuit)  

ARMATURE  LOSSES  AND  TEMPERATURE  RISE  (FULL  LOAD) 
40.  Length  mean  turn  of  armature  coil  (inches) 

n  = 
ln  = 

3 

9.1 
48.6 
20,500 

0.466 
3 
0.4 
9 

48.1 
20,700 

43  3 

41.  Ratio  of  active  to  total  copper 

0  337 

42.  Resistance  of  one  turn  (ohms) 

0  00132 

43.  Resistance  of  one  path  through  armature  (ohms) 

0  0564 

44.  Resistance  of  armature  (ohms) 

0  0141 

45.  IR  drop  in  armature  (volts)  

4.7 

46.  IR  drop  in  series  coils  (main  field)  (volts)  

1.6 

202          PRINCIPLES  OF  ELECTRICAL  DESIGN 

DESIGN  SHEET  FOR  CONTINUOUS-CURRENT  GENERATOR. — Continued 


Symbols 

Preliminary 
or  assumed 
values 

Final 
values 

47.  IR  drop  in  interpole  winding  
48.  IR  drop  (total)  at  brush-contact  surfaces  (volts)  .  .  . 
49.  I2R  loss  in  armature  winding  

2 
1  570 

2.25 
1  540 

50.  IZR  loss  to  be  radiated  from  armature  core  
51.  Full-load  useful  flux  per  pole 

4>  — 

530 

6  970  000 

52.  Flux  density  in  armature  core  below  teeth  

15  000 

53.  Full-load  apparent  tooth  density  (mean  value)  
54.  Radial  depth  of  armature  stampings  below  teeth 
(inches)  

Rd~ 

22,400 

4 

55.  Internal  diameter  of  armature  core  (inches)  
56.  Weight  of  iron  in  armature  core  below  teeth  (pounds) 
57.  Weight  of  iron  in  teeth  (pounds)  
58.  Iron  loss  in  core  (watts)  at  full  load  
59.  Iron  loss  in  teeth  (watts)  at  full  load.  
60.  Total  iron  loss  at  full  load  
61.  Total  watts  to  be  radiated  from  armature  core  
62.  Cooling  surface  of  active  belt  (square  inches)  

1,284 
390 
1,674 
2,204 

9.5 
428 
75 
1,284 
360 
1,644 

675 

63.  Cooling  surface  of  inner  bore  (square  inches)  

329 

64.  Cooling  surface  of  ducts  and  ends  (square  inches)  .  . 

1,820 

65.  Temperature  rise  of  armature  (degrees  Centigrade)  . 
STUDY  OF  FLUX  DISTRIBUTION  IN  AIR  GAP 

66.  Permeance  per  slot  pitch  at  center  of  pole  face  
67.  Equivalent  air  gap  (inches)  
68.  Drawing  of  pole  to  scale  and  measurement  of  flux 
paths  

T  = 
P\  = 

de  = 

Fig  78 

37 

98 
0.307 

69.  Permeance  curve  for  air  gap  
70.  Calculation  of  actual  tooth  densities  in  terms  of  air- 
gap  densities,  and  plotting  curve  connecting  these 
values. 
71.  Saturation  curves  for  air  gap,  teeth,  and  slots  for  a 
number  of  points  on  armature  surface  
72.  Open-circuit  m.m.f.  curve.  .  
73.  Open  circuit  flux  distribution  curve  A  
74.  Required  area  of  flux  curve  A  

Fig.  79 

Fig.  82 
Fig.  83 
Fig.  79 

106  3 

75.  Corrected  open-circuit  m.m.f.  curve  
76.  Corrected  flux  curve  A  to  check  with  required  area 
(item  74) 

Fig.  83 
Fig  84 

107 

77.   Maximum  value  of  armature  ampere-turns  per  pole 
(item  17)  . 

3  565 

78.  Resultant  m.m.f.  curve  to  get  flux  curve  B  

Fig.  83 

79.  Flux  curve  B.     Measured  area  =  

Fig.  84 

99.5 

80.   Required  area  of  full-load  flux  curve  C  = 

115 

81.  Estimated  additional  field  ampere-turns  to  bring  up 
area  of  flux  curve  from  area  of  B  curve  to  required 
area  of  C  curve  
82.   M.m.f.  curve  for  flux  curve  C  
83.  Full-load  flux  curve  C  

Fig.  83 
Fig.  84 

900 

PROCEDURE  IN  DESIGN  OF  D.C.  GENERATOR     203 

DESIGN  SHEET  FOR  CONTINUOUS-CURRENT  GENERATOR. — Continued 


COMMUTATOR  DESIGN 

Symbols 

Preliminary 
or  assumed 
values 

Final 
values 

84.   Diameter  of  commutator  (inches)  
85.  Peripheral  velocity  (feet  per  minute)  
86.  Volts  per  turn  of  armature  winding  (average  value). 
87.  Number  of  turns  between  bars  
88.  Total  number  of  commutator  bars  
89      Bar  pitch  (inch)                                             

^°  = 

13.5 
2,130 
2.87 
1 
171 
0.247 

90.  Thickness  of  mica  insulation  between  bars  (inch)  .  . 
91.  Width  of  bar  (on  surface)  (inch)  
92.   Radial  depth  of  bar  (inch)  
93.  Average  current  density    over  brush-contact  sur- 
face (amperes  per  square  inch)  
94.  Contact   area    of    brushes    (all  +  brushes    (square 
inches))                                 .    .                 

M  = 

35 
9.32 

0.032 
0.215 
1.75 

36.2 
9 

95.   Contact  area  per  brush  set  (square  inches)  
96.  Circumferential  width  of  brush  (inch)  
97.  Brush  width  referred  to  armature  surface  (inch)..  .  . 
98.  Total  axial  brush  length,  per  set  (inch)  
99.   Number  of  brushes  per  set  
100.  Axial  length  of  commutator  surface  (inches)  

CALCULATION  OF  FLUX  REQUIRED  IN  COMMUTATING  ZONE 

101.  End  flux  (maxwells)  
102.  Equivalent  slot  flux  (two  slots)  
103.  Total  flux  entering  teeth  in  commutating  zone.  .  .  . 
104.  Average  flux  density  of  commutating  field  
105.  Calculated  density  at  beginning  of  commutation  .  . 
106.  Calculated  density  at  end  of  commutation  .  . 

*.= 
*..= 

4.66 
7* 

4.5 
0.75 
1.083 
6 
4 
7* 

31,300 
11,730 
54,760 
712 

586 
838 

107.  Permissible   departure   from   ideal   values    of   flux 
density,  -f-  or  —  (gausses) 

1,720 

COMMUTATION  LOSSES  AND  TEMPERATURE  RISE 

108.  Brush  pressure,  pounds  per  square  inch  
109.   Resistance  per  square  inch  of  contact  surface  
110.  Total  brush  resistance  (full-load  conditions)  
111.  Total  voltage  drop  at  brush-contact  surfaces  
1  12.  J2/2  loss  (watts)  
113.  Friction  loss  (watts)  
114.  Total  commutator  loss  (watts)  

730 

1.5 
0.025 
0.00556 
2.25 
744 
324 
1,054 

115.  Total  cooling  surface  (square  inches)  
116.  Temperature  rise  of    commutator  (degrees  Centi- 
grade   

T 

"470.8 
48.4 

POLE  CORES  AND  FRAME 
117.  Leakage  coefficient  (assumed)  

1.2 

118.  Flux  density  in  pole  core  (full  load) 

16  500 

119.  Cross-sectional  area  of  pole  core  (square  inches) 

78  54 

120.  Pole-core  width                 \  ,. 
>  diameter  (inches) 

10 

121.  Pole-core  length  (axial)  / 
122.  Pole  length  (radial)  (inches)       

7 

123.  Flux  density  in  frame  (yoke  ring)  

15  000 

124.  Cross-sectional  area  of  frame  (square  inches)  

43.2 

204          PRINCIPLES  OF  ELECTRICAL  DESIGN 

DESIGN  SHEET  FOR  CONTINUOUS-CURRENT  GENERATOR. — Continued 


Symbols 

Preliminary 
or  assumed 
values 

Final 
values 

125.  Frame  width  (axial)  (inches)  

13 

126.  Frame  thickness  (at  center)  (inches)  

3.5 

128.  Calculation  ami  plotting  of  saturation  curve  for  the 
complete  magnetic  circuit  

FIELD  WINDINGS 

129.   SI  per  pole  for  total  magnetic  circuit  (no  load)  .  .  . 
130.  SI  per  pole  for  total  magnetic  circuit  (full  load)  .  . 
131.  SI   per   pole   to    compensate    for    distortion    and 
demagnetization 

Fig.  87 

5,930 
7,850 

400 

132.  SI  per  pole  in  shunt  field  at  full  load 

6  200 

133.  Thickness  of  shunt  winding  (inches) 

2 

2 

134.  Length  of  winding  space  for  shunt  coils  (inches)  .  . 
135.  Size  of  shunt  field  wire  (circular  mils) 

5 

4  880 

5 

5  178 

136.  Shunt  field  current  (full  load) 

4  56 

137.   Number  of  turns  per  pole  in  shunt  winding  
138.  SI  in  series  winding,  per  pole   .  .  . 

1,360 
1  650 

139.  Series  field  current.      ... 

326 

300 

140.   Number  of  turns  of  series  wire,  per  pole  
141.  Size  of  series  field  wire  (square  inches)  
142.  Resistance  of  series  field  (hot)  . 

0.25 

5H 

0  0028 

143.  Cooling  surface   of   field    coils    (one   pole)    (square 
inches)  

680 

144.  Surface   temperature   rise    of    field   coils    (degrees 
Centigrade)  .  . 

T 

43 

145.  Current  in  diverter.. 

30  56 

146.  Resistance  of  diverter.    . 

0  0275 

EFFICIENCY 

147.  Corrected  value  for  tooth  loss  at  full  load  
148.  Calculation  of  efficiency  at   M,  li,  H,  1,  and  1H, 
full-load  output  

360 

149.  Plotting  of  efficiency  curve  ... 

Fig    88 

63.  Numerical  Example — Calculations. — Items  (6)  and  (7): 
Number  of  Poles. — Refer  to  table  in  Art.  20  (page  81).  Usual 
number;  4  or  6.  On  page  78  in  same  article,  the  frequency  is 
stated  to  be  generally  between  10  and  40.  With  four  poles, 

r      4       600      .  .  .   , 

/  =  2  X  ^TTT,  which  is  satisfactory. 

Item  (8):  Ratio  of  Pole  Arc  to  Pole  Pitch.— Refer  Art.  19, 
pages  74  and  76.  Since  there  are  no  interpoles,  we  shall  make 
r  =  0.72. 

Item  (9) :  Specific  Loading. — Refer  Art.  19,  pages  74  and  76. 
For  75-kw.  machine,  try  q  =  475. 


PROCEDURE  IN  DESIGN  OF  D.C.  GENERATOR    205 

Item  (10):  Type  of  Winding.—  Refer  Art,  23,  page  84.  In 
this  case  some  doubt  exists  as  to  whether  a  wave  or  lap  winding 
should  be  adopted.  If  the  pressure  were  higher  —  say  500  volts  — 
a  series,  or  two-circuit,  winding  would  be  preferable.  With 
only  220  volts  to  be  generated,  we  shall  adopt  a  simplex  multiple 
winding  which,  with  our  four-pole  machine,  will  give  us  four 
armature  circuits  in  parallel. 

Item  (11):  Apparent  Air-gap  Density.  —  Refer  to  the  table  on 
page  75  in  Art.  19,  and  select  Bg  =  8,000. 

Items  (12)  and  (13):  Line  Current  =  7500%3o  =  326  amp. 
The  current  in  each  armature  conductor  will  be  one-quarter 
of  this  (Art.  23,  page  87)  if  we  neglect  the  shunt  exciting 
current.  The  shunt  excitation  of  a  75-kw.  machine  might 
amount  to  2.3  per  cent,  (see  Art.  58,  page  192),  so  that  the 

OOf! 

full-load  current  per  conductor  will  be  -j-  (1  +  0.023)  =  83.4 

amp.,  approximately. 

Item  (14):  Armature  Diameter.  —  Using  the  output  formula 
(43)  as  developed  in  Art.  19  (page  74),  we  have: 

_  75,000  X  60X10* 

tflj          6.45  X  7T2  X  8,000  X  475  X  0.72  X  600  " 

Referring  now  to  page  77,  we  can  use  formula  (46)  to  get  la 
in  terms  of  D.  The  ratio  la/rt  for  an  economical  design  of 
machine  of  this  size  and  number  of  poles,  will  probably  have  a 
value  between  0.5  and  0.8.  For  a  square  pole  face, 

t  =  rr.  .  r  .  o.72 

r          r 

and  this  seems  a  good  proportion  to  aim  at,  especially  as  it  will 
allow  of  cylindrical  pole  cores  being  used.  With  the  square 
pole  face, 


la  =          =  0.567D 
whence 


na  _.  _ 

"  0.567  '   7'5e 
and 

D  =  19.63  in. 

Let  us  therefore  decide  upon  armature  punchings  of  19.5  in. 
external  diameter. 


206          PRINCIPLES  OF  ELECTRICAL  DESIGN 

Item  (16):   Number  of  Inductors.  —  Refer   Art.   18,  page   72, 
and  Art.  19,  page  74. 

_       TrDq      TT  X  19.5  X  475  . 

Z  =  —  Y^  =  ~     —  ^5~i  —     -  =  350  (approximately) 

lc  OO.4 

Item  (17)  :  Full-load  Armature  Ampere-turns.  — 

Z/c  _  350  X  83.4 
(SI).  ==   2^  :        2X4  3'6c 

(18):  Lenpto  of  Air  Gap.—  Refer  Art.  36,  page  119. 

=  °-228;  or  (say)  *  in- 


Item  (19):  Maxwells  per  Pole.  —  Refer  formula  (38),  Chap.  IV, 
page  72. 

The  flux  per  pole  on  open  circuit,  if  Z  has  the  value  as  calculated 
for  item  (16),  is 

220  X  60  X  4X  108 
*  -       4X600X350        =  6'290'000  maxwells' 

Item  (20)  :  Pole  Pitch.—  Refer  Art.  19,  page  74  and  Art.  20, 
page  78. 

7TX19.5       ,_-••. 
T  =  --  £  --  =  15.34  in. 

Item  (21)  :  Pole  Arc.—  Refer  Art.  19,  page  74. 

r  X  r  =  15.34  X  0.72  =  11.05,  or  (say)  11  in. 
Items  (22),  (23)  and  (24):  Dimensions  of  Air  Gap.— 


*  =  =  786  gq  cm 

Bg          8,000      =  122  sq.  in. 

whence  la  =  12Ki  =  11.1,  and  the  axial  length  of  pole  face  will 
be  something  less,  or  (say)  10.6,  to  avoid  the  large  amount  of 
flux  which  would  otherwise  curve  round  into  the  flat  surface  of 
the  core  discs,  where  it  would  cause  eddy  currents.  These  axial 
dimensions,  both  of  armature  and  pole  shoe,  are,  however,  sub- 
ject to  correction  after  the  actual  winding  details  have  been 
settled;  because  the  practical  considerations  may  lead  to  a 
change  in  the  number  of  inductors  (Z),  and  a  corresponding 
change  in  the  amount  of  flux  entering  the  armature. 


PROCEDURE  IN  DESIGN  OF  D.C.  GENERATOR       207 

Item  (25):  Cross-section  of  Armature  Conductors.  —  By  formula 
(51),  page  97. 


QQ    \ 

whence  area  of  cross-section  =  0    '7.  =  0.0402  sq.  in. 

Z,(Ji  O 

Items  (26)  to  (31):  Conductor  and  Slot  Dimensions.  —  It  is  neces- 
sary to  find  by  trial  the  best  arrangement  of  slots  and  conductors 
to  provide  approximately  350  inductors  (item  (16)).  The  number 
of  slots  per  pole  should  not  be  less  than  10  (Art.  23,  page  84 
and  Art.  26,  page  93),  and  there  must  be  an  even  number  of 
conductors  in  each  slot.  A  winding  consisting  of  44  or  45  slots, 
each  with  eight  conductors,  would  be  a  possible  arrangement; 
but  the  slot  pitch  would  be  large  with  so  few  slots.  It  would 
seem  advisable  to  have  a  winding  with  six  conductors  per  slot. 
The  number  of  slots  would  then  be  approximately,  35%  =  58.3. 
Either  14  or  15  slots  per  pole  would  be  suitable  for  a  parallel 
winding;  but  since  it  is  usual  to  provide  the  armature  punchings 
of  four-pole  machines  with  an  uneven  number  of  slots,  so  that 
the  armature  core  can  be  used  for  a  two-circuit  winding,  we 
shall  adopt  a  winding  of  57  slots  with  six  conductors  per  slot, 
making  the  corrected  value  of  Z  =  57  X  6  =  342 

The  slot  pitch  (refer  Art.  25,  page  92),  is, 


Oi 

The  number  of  teeth  between  pole  tips  is, 
15.34  -  11 


1.076 


=  4.03 


Had  this  figure  been  less  than  3.5  (see  Art.  26,  page  93),  it 
might  have  been  advisable  to  increase  the  number  of  teeth,  or 
widen  the  space  between  pole  tips. 

In  order  to  determine  the  actual  dimensions  of  the  armature 
conductors,  it  will  be  found  convenient  to  assume,  a  width  of  slot. 
This  should  be  about  one-half  the  slot  pitch,  or,  say,  0.5  in. 
(see  Art.  25,  page  92).  If  we  adopt  the  arrangement  of  con- 
ductors in  the  slot  shown  in  Fig.  77,  the  width  of  each  conductor 
will  be  one-third  of  the  total  width  available  for  copper.  The 
cotton  covering  on  each  conductor  would  add,  say,  16  mils  total 
to  its  thickness;  and  the  slot  insulation  will  be  about  0.035  in. 


208 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


thick  (see  Art,  28,  page  96).  The  space  left  for  copper  is 
therefore  0.5  -  (0.07  +  0.048)  =  0.382;  and  the  width  of  each 
conductor  is  one- third  of  this  amount,  or  0.127.  Let  us  make 
this  J£  in.  (0.125).  The  depth  of  the  (rectangular)  conductor 

0  0402 

=  0.322,  or,  say,  %6  in.  (0.312). 


will  be 


These  are  the 


0.125 

dimensions  called  for  under  item  (26);  and  the  corrected  value 
for  item  (25)  is  0.312  X  0.125  =  0.039  sq.  in. 
The  required  slot  depth  is  made  up  as  follows: 

Hard-wood  wedge,  which  should  be  about 0.200  in. 

Insulation  above,  below,  and  between  the  coils 

=  3  X  0.035 0.105  in. 

Cotton  covering  on  wires  (twice  0.016) 0.032  in. 

Copper .  0.624  in. 


0.961  in. 
or,  say,  1  in.  for  the  dimension  d  in  Fig.  77. 

Before  finally  adopting  these  dimensions,  it  will  be  necessary 

to  see  that  the  flux  density 
in  the  teeth  is  not  excessive 
(item  (39)). 

Items  (32)  to  (34) :  Tooth  Di- 
mensions.— The  width  of  tooth 
at  the  top  is  t  =  X  —  s  =  0.576. 

The  circumference  of  the 
circle  through  the  bottom  of 
the  slots  is  TT  X  17.5;  and  since 
the  slots  have  parallel  sides,  the 
width  of  tooth  at  the  root  is 
TT  X  17.5 


FIG.  77. — Arrangement  of  conduc- 
tors in  slot 


57 


-  0.5  =  0.466. 


Items  (35)  and  (36) :  Cooling  Duds.— Refer  Art.  33,  page  105. 
Not  more  than  three  ducts  should  be  necessary  in  an  armature 
11  in.  long.     Each  duct  might  be  0.4  in.  wide. 
Item  (37):  Net  Length  of  Armature.— Refer  Art.  31,  page  103. 

ln  =  0.92  (11.1  -  1.2)  =  9.1  in. 

Item  (38) :  Net  Cross-section  of  Teeth  under  Pole. — The  cross- 
section  of  iron  in  the  teeth  under  one  pole,  at  a  point  halfway 
up  the  tooth,  is, 

9.1  X  0.521  X        X  0.72  =  48.6  sq.  in. 


PROCEDURE  IN  DESIGN  OF  D.C.  GENERATOR    209 

Item  (39):  Flux  Density  in  Teeth.—  Refer  Art.  31,  page  102, 
and  Art.  32,  page  104.  Before  calculating  the  flux  density  in 
the  teeth,  it  is  necessary  to  correct  the  figure  for  flux  per  pole 
(item  (19)),  because  the  number  of  face  conductors  (Z)  has  been 

changed.     Thus, 

350 
$  =  6,290,000  X          =  M30,000  maxwells. 


The  apparent  flux  density  at  the  center  of  the  tooth,  under 
open-circuit  conditions,  is  therefore, 


=  20,500  gausses. 

Referring  to  the  table  on  page  104,  it  will  be  seen  that  a  maximum 
tooth  density  of  22,000  is  permissible  when  the  frequency  is  20. 
We  do  not  yet  know  what  will  be  the  actual  flux  density  at  the 
root  of  the  teeth  under  full-load  conditions;  but  it  is  not  likely 
to  be  excessive,  and  we  may  proceed  with  the  design. 

At  this  stage  it  might  be  well  to  alter  the  gross  length  of  the 
armature  core  from  11.1  to  exactly  11  in.,  reducing  the  net 
length  accordingly.  This  will  account  for  the  corrected  values 
of  items  (37)  to  (39)  in  the  last  column  of  figures  of  the  design 
sheets. 

Item  (40):  Length  per  Turn  of  Armature  Coil.  —  Referring  to 
Art.  30,  page  97,  we  have, 

1.15s       1.15X0.5 
sma=    ~X~       ~L076~ 
whence 

a  =  32°  20' 
and 

cos  a  =  0.845 


By  formula  (52),  page  98, 

_  2  X  15.34 
le  ~       0.845 


+  4  +  3  =  43.3  in. 


Item  (41) :  Ratio  of  Copper  in  Slots  to  Total  Armature  Copper. — 
Refer  Art.  34,  page  109. 

2la  22 


2la  +  le      22  +  43.3 


=  0.337 


Items  (42)  to  (45) :  Armature  Resistance. — Refer  Art.  30,  page 
97.     The  total  length  of  one  turn  of  the  armature  winding  is 


14 


210          PRINCIPLES  OF  ELECTRICAL  DESIGN 

65.3  in.,  and  by  formula  (21)  page  83,  the  resistance  at  about 
60°C.  will  be 

&.  Pv  Q 

R  =  -  —7  =  0.00132  ohm. 

0.039  X  106  X  - 

7T 

There  are  Z/2  or  171  turns  in  the  armature  winding,  and  there- 
fore 171/4  =  42.75  turns  in  series  in  each  armature  circuit. 
The  value  of  item  (43)  is  therefore  42.75  X  0.00132  =  0.0564 
ohm;  and  of  item  (44),  one-quarter  of  this  amount,  or  0.0141 
ohm.  The  IR  drop  in  armature  winding  is  0.0564  X  83.4  =  4.7 
volts,  or  2.04  per  cent,  of  the  full-load  terminal  voltage.  This 
compares  favorably  with  the  approximate  figures  given  on 
page  99. 

Item  (46)  :  Pressure  Drop  in  Series  Winding.  —  Refer  Art.  (43) 
page  139.  We  may  assume  this  voltage  drop  to  be  one-third 
of  4.7  or,  say,  1.6  volts. 

Item  (48):  Pressure  Drop  at  Brushes.  —  Refer  Art.  53,  page 
179.  Assume  two  volts. 

Items  (49)  and  (50)  :  Watts  Lost  in  Armature  Windings.  —  Total 
PR  =  El  =  4.7  (83.4  X  4)  =  1,570  watts.  Item  (50)  is  the 
portion  of  this  total  loss  which  occurs  in  the  "active"  copper 
of  the  armature;  its  value  is, 

1570  X  0.337  =  530  watts, 

wherein  the  factor  0.337  is  item  (41)  of  the  design  sheets. 

Item  (51)  :  Flux  Entering  Armature  at  Full  Load.  —  Refer  Art. 
43,  page  135. 

The  volts  to  be  developed  at  full  load  are, 

230  +  4.7  +  1.6  +  2  =  238.3 
The  full-load  flux  must  therefore  be, 

238  3 
6,430,000  X  --  =  6,970,000  maxwells. 


Items  (52)  to  (55):  Flux  Density  in  Armature  Core.  —  Internal 
Diameter.  —  Usual  flux  densities  for  different  frequencies  are 
given  in  the  table  in  Art.  32  (page  104).  A  density  of  14,000 
gausses  would  be  satisfactory;  but  since  the  losses  in  the  teeth 
are  likely  to  be  below  the  average  —  because  a  1-in.  depth  of 
slot  is  small  for  a  machine  of  this  size  —  a  density  of  15,000  may 


PROCEDURE  IN  DESIGN  OF  D.C.  GENERATOR      211 

be  tried.  Bearing  in  mind  that  the  maximum  flux  in  the  arma- 
ture core  is  one-half  of  the  total  flux  per  pole,  we  have, 

3> 
RdXlnX  6.45  X  15,000  =  ^ 

whence 

6,970,000 

d  ~  2  X  9  X  6.45  X  15,000  ~ 

Item  (56):  Weight  of  Iron  in  Core. — The  weight  of  a  cubic 
inch  of  iron  is  0.28  Ib.  and  the  total  weight  of  iron  in  the  core 
below  the  teeth  will  therefore  be 

0.28  X  9  X  ^[(17.5)2  -  (9.5)2]  ==  428  Ib. 

Item  (57) :  Weight  of  Iron  in  Teeth.— 

0.28  X  1  X  0.521  X  9  X  57  =  75  Ib. 

Items  (58)  to  (60):  Iron  Losses.— Refer  Art.  31.  The  watts 
per  pound  are  read  off  the  curve  of  Fig.  34  on  page  102;  thus, 
for  the  armature  core  we  have, 

Bf        15,000  X  20 
1,000  1,000 

whence  watts  per  pound  =  3;  and  total  watts  =  3  X  428  = 
1,284. 

Similarly,  for  the  teeth  (item  (59))  we  get  a  loss — with  full- 
load  flux — of  390  watts.  The  total  iron  loss  of  1,674  watts,  being 
2.23  per  cent,  of  the  output,  is  rather  higher  than  the  average 
as  given  in  the  table  on  page  104;  but  if  the  temperature  rise 
is  not  excessive,  it  will  not  be  necessary  to  reduce  the  flux 
densities. 

Item  (61):  Total  Loss  to  be  Radiated  from  Armature  Core. — 
Refer  Art.  34,  page  109.  The  copper  loss  to  be  added  to  the 
total  iron  loss  is  the  amount  of  item  (50). 

Items  (62  to  (64):  Cooling  Surfaces.— Refer  Art.  34,  page  107. 

Cooling  surface  item  (62)  =  w  X  19.5  X  11  =  675  sq.  in. 
Cooling  surface  item  (63)  =  w  X    9.5  X  11  =  329  sq.  in. 

Cooling  surface  item  (64)  =  ^  [(19.5)2  -  (9.5)2]  X  4  =  1,820 
sq.  in. 


212          PRINCIPLES  OF  ELECTRICAL  DESIGN 

Item  (65):  Temperature  Rise.  —  Refer  Art.  34,  page  107.  The 
radiating  coefficient  for  the  cylindrical  surfaces  is,  by  formula 
(54), 

i  50Q   I    Q  070 
For  the  outside  surface  wc  =  -  -  —  0.0457 

lUUjUUU 

v     ^    -     -j  1,500  +  1,500 

For  the  inside  surface  wc  =  --  '  —  =  0.03 

1UU,UUU 

In  the  case  of  the  ducts,  it  should  be  noted  that  the  radial 
depth  of  the  armature  stampings  is  large  because  of  the  wide 
polar  pitch,  and  instead  of  taking  the  velocity  of  air  through 
the  vent  ducts  as  one-tenth  of  the  outside  peripheral  velocity, 

we  shall  assume  a  lower  value,  making  vd  =  r^-     Thus,  the  cool- 

1.4 

ing  coefficient  for  the  ducts  and  ends  will  be,  by  formula  (56), 


The  procedure  is  exactly  as  followed  in  the  example  on  page  111, 
and  the  temperature  rise  is  found  to  be  37°C.,  which  is  within 
the  specified  limit  of  40°. 

Items  (66)  and  (67):  Equivalent  Air  Gap.—  Refer  Art.  36, 
page  117. 

The  permeance  of  the  air  gap  over  one  slot  pitch  at  the  center 
of  the  pole  face  where  the  actual  clearance  is  5  =  Y±  in.  may  be 
written, 


=  98 
and  the  equivalent  air  gap,  as  given  by  formula  (58),  will  be 


6.45  X  11  X  1.076       ._ 

-98~        -  =  0-78  cm. 

=  0.307  in. 

Item  (68):  Drawing  of  Equivalent  Flux  Lines. — Refer  Art.  41, 
page  129. 

Before  completing  the  drawing  of  the  pole  shoe  as  shown  in  Fig. 
78,  it  is  well  to  estimate  the  cross-section  of  the  pole  core,  as 


PROCEDURE  IN  DESIGN  OF  D.C.  GENERATOR    213 

this  will  be  helpful  in  deciding  upon  the  most  suitable  shape  of 
the  pole  shoe.  The  full-load  flux  per  pole  (item  (51))  is  6,970,000 
maxwells;  and  if  we  assume  a  leakage  coefficient  of  1.2  (Art. 
56,  page  187),  the  flux  to  be  carried  by  the  pole  core  is  8,370,000 
maxwells,  approximately.  The  density  in  the  core  may  be  as 
high  as  16,000  gausses,  and  the  cross-section  is  therefore 
8,370,000 


16,000  X  6.45 
cross-section, 


=  81  sq.  in.     If  the  pole  core  is  made  of  circular 


Diameter  =  -\|81  X      =  10.15 
=  (say)  10  in. 

A  laminated  pole  shoe  of  the  shape  shown  in  Fig.  78  will  be 
suitable.     The  stampings  would  be  riveted  together   and    at- 


FIG.  78. — Graphical  construction  for  calculating  permeance  of  flux  paths. 

tached  by  screws  to  the  face  of  the  cylindrical  pole  core.  The 
distance  aO,  measured  on  the  armature  surface,  is  one-half 
of  item  (21),  or  5.5  in.;  while  al  is  one-half  of  the  pole  pitch  r 
(item  (20)),  and  measures  7.67  in.  The  tip  of  the  pole  is  shaped 
so  that  the  air  gap  increases  from  the  point  /  outward,  being  }/% 
in.  greater  at  the  ends  of  the  pole  arc  than  at  the  center.  The 
extreme  point  of  the  pole  shoe  is  rounded  off*  with  a  %Q-m. 
radius.  The  reference  points  on  the  armature  surface  have  been 
chosen  for  convenience  at  intervals  of  10°,  except  in  the  neighbor- 
hood of  the  pole  tip,  where  the  selected  points  are  only  5°  apart. 


214 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


The  construction  of  the  flux  lines  is  as  explained  in  Art.  41, 
and  the  measurements  taken  off  the  drawing  have  the  following 
values : 


Points  on  armature 

Length  of  equivalent  flux 
line  =  I  cm. 

Permeance  per  square  centi- 
meter =  I/I 

a,  b,  c,  d,  e,  f 

0.78 

1.28 

9 

0.93 

1.075 

h 

1.17 

0.855 

i 

2.03 

0.492 

3 

4.13 

0.242 

k 

7.12 

0.141 

I 

13.1 

0.076 

m 

20.1 

0.05 

FIG.  79. — Curves  of  air  gap  permeance  and  open  circuit  flux  distribution- 

Item  (69):  Permeance  Curve. — Refer  Art.  39.  The  curve 
marked  P  in  Fig.  79  has  been  plotted  from  the  above  figures. 
It  shows  the  variation  of  air-gap  permeance  between  pole  and 
armature  at  all  points  from  the  center  of  pole  face  to  a  point, 
m,  10°  beyond  the  geometric  neutral. 

Item  (70):  Actual  Tooth  Densities.— Refer  Art.  37,  page  119. 
In  order  to  make  use  of  formula  (62),  giving  the  relation  between 


PROCEDURE  IN  DESIGN  OF  D.C.  GENERATOR    215 


Bg  and  Bt  at  high  densities,  it  will  be  found  convenient  to  prepare 
a  table  similar  to  the  one  below. 


Bt 

TT 

d  +  MS 

M(rf  +  5) 

tig 

21,000 

450 

46.7 

0.217 

10,360 

22,000 

670 

33.0 

0.224 

10,950 

23,000 

950 

24.2 

0.233 

11,600 

24,000 

1,360 

17.6 

0.245 

12,300 

25,000 

2,000 

12.5 

0.264 

13,100 

Values  of  the  actual  tooth  density,  Btj  from  21,000  to,  say, 
25,000  gausses  are  assumed,  and  the  corresponding  values  of  H 


^^r**" 

+^ 

**  

—  *" 

24000 

^ 

^^ 

-  <"• 

«*a 

^x- 

^^^ 

&3 

II      9QQOO 

^ 

^ 

01 

] 

^ 

X1 

3 
O   29000 

/ 

/ 

Jj 

/ 

/ 

•3 

C 
<u    21000 

/ 

/ 

Q   *.iuw 
X 

/ 

/ 

£ 

20000 

/ 

IQOOfl 

/ 

100  200  300  400    500  600  700  800  900  1000        1200         1400         1600         1800        2000 
Magnetizing  Force   (Gilberts  per  Centimeter)  =  H 

FIG.  80. — B-H  Curve  for  armature  stampings — high  values  of 
magnetization. 

and  M  are  then  found  by  referring  to  the  B-H  curve,  Fig.  80, 
which  is  similar  to  Fig.  4  except  that  the  quantities  concerned 
are  expressed  as  B  and  H  because  this  is  more  convenient  for 
obtaining  /z. 

The  quantity  which  is  a  function  of  ju>  in  formula  (62),  can 
now  be  determined,  and  the  corresponding  values  of  B0  readily 
calculated. 

In  this  example,  we  shall  consider  the  tooth  density  in.  the 


216 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


root,  or  narrowest  part,  of  the  tooth;  and  t  in  the  formula  is 
therefore  taken  as  0.466  in.  (item  34).  The  results  of  this 
calculation  are  shown  graphically  in  the  upper  dotted  curve 
of  Fig.  81. 


Tooth  Density,  JBj  ,  (  Gausses  ) 

/ 

> 

// 

// 

/ 

/ 

/ 

A 

/ 

F 

jrmi 

la(( 

2) 

I 

/ 

I 

/ 

API 

arer 

Formula  (63)    L 
t  Tooth  Density! 

1  / 

1 

[/ 

I/ 

1 

f 

f 

/ 

/ 

i 

/ 

1 

2000300040005000600070008000        10000       12000      14000 
Air  Gap  Density,  B  g,  (  Gausses ) 

FIG.  81. — Curve  giving  relation  between  air  gap  and  tooth  densities. 

The  "apparent"  tooth  density  at  the  bottom  of  the  tooth  is, 
by  formula  (63), 

1.076  X  ir 


which  enables  us  to  plot  the  lower  dotted  curve  of  Fig.  81.  The 
actual  density  in  the  iron  of  the  teeth  is  almost  exactly  expressed, 
for  the  low  values,  by  formula  (63);  while  at  very  high  densities, 
the  actual  tooth  density  approaches  more  and  more  nearly  the 
values  calculated  by  formula  (62)  without  ever  quite  reaching 
them.  It  is  therefore  possible  to  draw  a  curve,  such  as  the  full 


PROCEDURE  IN  DESIGN  OF  D.C.  GENERATOR    217 

line  in  Fig.  81,  which  very  closely  expresses  the  true  relation 
between  the  tooth  density  and  the  average  density  over  the  slot 
pitch,  for  the  entire  range  of  values  from  zero  to  the  highest 
attainable. 

Item  (71):  Saturation  Curves  for  Air  Gap,  Teeth,  and  Slots. — 
Refer  Art.  38,  page  121,  and  Art.  42,  page  132.  We  are  now  in 
a  position  to  plot  curves  similar  to- those  of  Fig.  49,  page  133; 
and,  in  order  to  obtain  a  proper  value  for  the  ampere-turns  neces- 
sary to  overcome  the  reluctance  of  the  teeth,  the  correction  for 
the  taper  of  teeth  should  be  applied.  The  results  of  the  calcu- 
lations for  the  teeth  are  shown  in  tabular  form;  the  meaning  of 
the  different  columns  of  figures  being  as  follows: 

First  column:  Assumed  values  of  air-gap  density  Bg,  including 
the  highest  value  likely  to  be  attained. 

Second  column:  The  corresponding  values  of  the  density  Bt 
at  the  bottom  of  the  tooth  (read  off  the  full-line  curve  of  Fig.  81). 

Third  column:  The  magnetizing  force  H,  calculated,  when 
necessary,  by  applying  SIMPSON'S  rule  (Formula  64),  as  explained 
in  Art.  38. 

Fourth  column:  The  ampere-turns  required  to  overcome  the 
reluctance  of  the  teeth,  being 

Hde  X  2.54 

0.47T 

where  de  is  the  " equivalent"  length  of  tooth;  its  numerical  value, 
in  this  example,  being  (1  -f  0.25)  -  0.307  =  0.943  in. 


B0  . 

Bt 

H 

(SI)t 

12,000 

24,400 

790 

1,500 

10,000 

22,100 

347 

642 

8,000 

19,300 

110 

210 

6,000 

15,800 

20 

38 

As  an  example  of  the  method  of  calculation,  consider  the  value 
Bg  =  10,000;  the  corresponding  value  of  tooth  density,  as  read 
off  Fig.  81,  is  Bt  =  22,100.  This  is  the  actual  density  at  the 
root  of  the  tooth.  Referring  to  items  (32)  and  (34),  it  is  seen 
that,  over  a  distance  of  1  in.,  the  width  of  tooth  changes  by  the 
amount  0.576  -  0.466  =  0.11  in.  The  width  of  tooth  at  the 
distance  de  from  the  bottom  of  tooth  (see  Fig.  38)  is  therefore 


218          PRINCIPLES  OF  ELECTRICAL  DESIGN 
0.466  +  (0.11  X  0.943)  =  0.57;  and  the  density  at  this  point  is 
Bw  =  22,100  X  =  18,100 


At  the  halfway  section,  Bm  =  —  --  5L±i  =  20,100 

The  corresponding  values  of  H,  as  read  off  the  B-H  curves,  Fig.  80 
and  Fig.  2,  are: 

At  bottom,  Hn  =  700 

At  middle,  Hm  =  310 

At  top,  Hw  =  144. 
By  formula  (64),  we  have, 

„       700       2  X  310   ,    144 
Average  H  --    -g-  H  ---  ^  --  +  -^-  =  347 

which  is  appreciably  higher  than  the  value  of  H  at  the  section 
halfway  between  the  two  extremes.  This  difference  will,  how- 
ever, hardly  be  noticeable  on  low  values  of  tooth  density;  and 
indeed  the  somewhat  tedious  work  involved  in  the  above  calcu- 
lation is  quite  unnecessary  with  small.  values  of  tooth  density, 
because  the  ampere-turns  required  to  overcome  tooth  reluctance 
are  then,  in  any  case,  but  a  small  percentage  of  the  air-gap 
ampere-turns.  The  values  of  H,  in  the  above  table,  for  Bg  = 
8,000  and  Bg  =  6,000,  are  those  corresponding  to  the  average 
values  of  the  tooth  density  (Bm). 

Having  plotted  in  Fig.  82  the  curve  for  the  teeth  only,  the 
straight  line  for  the  air-gap  proper  can  now  be  drawn  for  the 
points  under  the  center  of  the  pole  face  where  the  equivalent 
air  gap  is  de  =  0.307  in.  Obviously,  since  HI  =  0.47r>S/,  and 
B  has  the  same  value  as  H  in  air,  we  may  write, 


=  0.62B, 

This  gives  us  the  line  marked  A  in  Fig.  82.  The  addition  to  this 
curve  of  the  ampere-turns  for  the  teeth  and  slots,  results  in  the 
curve  marked  a,  6,  c,  d,  e,  f,  which  may  be  used  for  all  points 
under  the  pole  where  the  permeance  has  the  same  value  as  at  a. 
The  curves  for  the  other  points  on  the  armature  surface  may  now 
be  drawn  as  explained  in  Art.  42. 

Items  (72)  to  (76)  :  Flux  Distribution  on  Open  Circuit.  —  Refer 
Arts.  40,  41,  and  42.     From  the  point  R  on  the  permeance  curve 


PROCEDURE  IN  DESIGN  OF  D.C.  GENERATOR    219 

(Fig.  79)  draw  a  straight  line  to  the  point  on  the  datum  line 
immediately  below  the  pole  tip,  in  order  to  obtain  the  curve  A 
of  flux  distribution  on  open  circuit,  all  as  explained  on  page 
131  (Art.  41).  Measure  the  area  of  this  flux  curve,  and  draw 


11000 


10000 


0  1000         2000          3000         4000         5000          6000         7000         8000 

Ampere -Turns  per  Pole 

FIG.  82. — Saturation  curves  for  air-gap,  teeth,  and  slots. 


the  dotted  rectangle  of  equal  area.     The  height  of  this  rectangle 
represents  the  average  air-gap  density  over  the  pole  pitch,  and 

•     i  u    a     /  x  6,430,000 

its    numerical    value    is    Bg    (average)  =  ^  ^  x  15  34  X  11  = 

5,910  gausses. 

Using  this  length  as  a  scale  for  measuring  other  ordinates  of  the 


220 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


flux  curve,  the  value  of  Bg  at  all  points  on  the  armature  periphery 
can  be  determined. 

The  ampere-turns  between  pole  and  armature,  due  to  field 
excitation  on  open  circuit,  have  at  every  point  on  the  armature 


surface  the  value  SI  = 


Bf. 


0.47T  X  permeance  per  square  centimeter 


9000 


M.M.F.  Curves 

Ampere  Turns  per  Pole 

Field  M.M.F.  (Open  Circuit) 

\  Field  M.M.F.  (Full  Load) 


I     kj  ihg   f    e    d    c     b    a,    b    c    d    e    f    ghijk    I    m 

Reference  Points 

FIG.  83. — Curves  showing  distribution  of  m.m.f.  between  pole  and 

armature. 

The  actual  figures  are  given  in  the  following  table. 


Point  on  armature  surface 

P,q.  cm. 

Bg 

SI 

o,  6,  c,  d,  e,  /, 

1.28 

7,840 

4,860 

9 

1.075 

6,580 

4,860 

h 

0.855 

5,220 

4,850 

i 

0.492 

2,910 

4,700 

J 

0.242 

I7305v 

4,275 

k 

0.141 

581 

3,270 

I 

0.076 

0 

0 

From  these  figures  the  dotted  m.m.f.  curve  of  Fig.  83  has  been 
plotted.  Observe  now  that  the  m.m.f.  represented  by  4,860 
ampere-turns  per  pole  is  not  sufficient  to  overcome  the  reluctance 
of  the  teeth,  and  in  order  to  obtain  the  required  total  flux  on 


PROCEDURE  IN  DESIGN  OF  D.C.  GENERATOR    221 


open  circuit,  the  m.m.f.  between  pole  face  and  armature  core, 
(i.e.,  bottom  of  slots)  must  exceed  this  value.  Referring  again 
to  Fig.  82,  it  will  be  seen  that,  for  a  density  of  7,840  gausses  under 
the  pole  face,  the  m.m.f.  to  overcome  tooth  reluctance  amounts 
to  a  little  over  200  ampere-turns.  The  ordinates  of  the  dotted 
m.m.f.  curve  of  Fig.  83  may  therefore  be  increased  throughout 
in  the  proper  proportion,  the  maximum  addition  being  200 
ampere-turns.  This  corrected  curve  may  now  be  used  to  plot — 
with  the  aid  of  the  magnetization  curves  of  Fig.  82 — the  actual 
distribution  of  flux  over  the  armature  surface,  when  the  effect 


12000 
11000 
10000 

£ 

^ 

^ 

^ 

•^ 

^ 

^ 

'\ 

\ 

soon 

^ 

^ 

^ 

^ 

B 

\\ 

A/ 

/" 

? 

-X 

^ 

N 

\\ 

6000 
5000 

/ 

/ 

/ 

^ 

^ 

\   u 

1 

/ 

,/ 

^ 

\ 

/ 

/ 

/ 

I 

/ 

/ 

\ 

90OT1 

/ 

// 

l\ 

// 

/ 

H 

0 
1000 
2000 
3000 
4000 
5000 

^ 

y/ 

^ 

^> 

'] 

AH 

siti 

m 

Direction  of  Travel  of  Conductors 

^ 

s 

V 

Flux  Curves 
(  Distribution  of  Flux  Density 
over  Armature  Surface  ) 

\ 

x 

I   kjihg  f    e    d    c    b    a  b    c   d   e   f   ghijk   I    m 

Reference  Points 

FIG.  84. — Curves  of  flux  distribution  over  armature  surface. 

of  tooth  saturation  is  taken  into  account.  This  has  been  done 
in  Fig.  84,  where  the  curve  marked  A  is  similar  to  the  flux  curve 
of  Fig.  79  except  in  so  far  as  its  shape  may  be  modified  by  tooth 
saturation. 

The  procedure  above  described  for  obtaining  the  actual  flux 
distribution  curve  is  logical  and  correct;  but  for  practical  pur- 
poses it  is  usually  permissible  to  assume  that  the  curve  A  of 
Fig.  79  shows  the  actual  flux  distribution,  the  slight  modifica- 
tion brought  about  by  variable  degrees  of  tooth  saturation  being 
neglected.  It  is  then  a  simple  matter  to  plot  the  required  open 
circuit  field  m.m.f.  curve  directly  by  taking  from  Fig.  82  the 


222          PRINCIPLES  OF  ELECTRICAL  DESIGN 

values  of  SI  corresponding  to  each  known  value  of  the  air-gap 
density,  Bg. 

The  average  ordinate  of  the  curve  A  in  Fig.  84  —  as  obtained 
by  dividing  the  area  under  the  curve  by  the  length  of  the  base  — 
is  found  to  check  within  1  per  cent,  of  the  required  amount 
(average  Bg  —  5,910  gausses).  Had  there  been  an  appreciable 
difference  between  the  calculated  and  measured  areas  of  the 
flux  curve,  it  would  have  been  necessary  to  correct  the  m.m.f 
curve  of  Fig.  83,  and  re-plot  the  flux  curve  A  of  Fig.  84. 

Items  (78)  and  (79):  Flux  Distribution  under  Load.  —  Refer 
Art.  43,  page  137.  The  curve  of  armature  m.m.f.,  of  which  the 
maximum  value  is  3,565  ampere-turns  (item  (17)),  may  now  be 
drawn  in  Fig.  83.  The  point  k  has  been  selected  for  the  brush 
position  because  the  (positive)  field  m.m.f.  has  here  about  the 
same  value  as  the  (negative)  armature  m.m.f.  From  the  re- 
sultant m.m.f.  curve,  the  flux  curve  B  of  Fig.  84  is  plotted,  the 
area  of  which  —  measured  between  brush  and  brush  —  is  found 
to  be  99.5  sq.  cm.  while  curve  A  measures  107  sq.  cm.  The 
average  air-gap  density  is  therefore  less  than  before  current 
was  taken  from  the  armature,  and  the  loss  of  flux  is  due  partly 
to  distortion  (tooth  saturation)  and  partly  to  the  demagnetizing 
ampere-turns  (brush  shift). 

Items  (80)  to  (83):  Corrected  Full-load  Flux  Distribution.— 
Refer  Art.  43.  The  e.m.f.  to  be  developed  at  full  load  is  238.3 
volts,  obtained  by  adding  the  numerical  values  of  items  (45), 
(46),  and  (48),  to  the  full-load  terminal  voltage.  The  final  flux 

1  0fi  3  ^  238 
curve  C  should  therefore  have  an  area  of  - 


sq.  cm.;  the  number  106.3  being  item  (74).  In  order  to  estimate 
the  probable  increase  in  field  excitation  to  obtain  this  increase 
of  flux,  we  may  follow  the  method  outlined  on  page  138.  The 
ampere-turns  necessary  to  bring  up  the  flux  from  the  reduced 
value  under  curve  B  to  the  original  value  under  curve  A}  are 
calculated  by  assuming  that  the  air-gap  density  under  the  pole 

99  5 
has  changed  from  7,800  gausses  to  7,800  X  ^y  =  7,260  gausses; 

and  the  ampere-turns  necessary  to  increase  the  developed  volts 
from  220  to^  238.3  are  calculated  by  assuming  that  the  air-gap 
density  under  the  pole  must  be  raised  from  7,800  to  7,800  X 

238  3 

=  8,450  gausses.     The  total  additional  excitation  is  indi- 


cated by  the  distance  SS'  in  Fig.  82,  its  value  being  about  900 


PROCEDURE  IN  DESIGN  OF  D.C.  GENERATOR    223 

ampere-turns.  This  must  be  added  to  the  open-circuit  m.m.f. 
curve  of  Fig.  83,  all  the  ordinates  of  which  must  be  increased  in 

.     5,050  +  900      _,  . 

the  ratio  -     ,  n,n .     The  new  resultant  m.m.f.  curve — ob- 

o,uoU 

tained  by  adding  this  full-load  field  m.m.f.  to  the  armature 
m.m.f. — may  now  be  used  to  plot  the  final  full-load  flux  curve 
C  of  Fig.  84. 

Items  (84)  and  (85) :  Diameter  of  Commutator. — Refer  Art.  53, 
page  181. 

A  diameter  of  commutator  not  exceeding  three-quarters  of 
the  armature  diameter  will  be  suitable.  Let  us  try  a  diameter 

13  5 

Dc=  13.5  in.,  making  vc  =  3,070  X  -  ;r^  =  2,130ft.  per  minute. 

iy.o 

This  dimension  is  subject  to  correction  if  the  thickness  of  the 
individual  bar  does  not  work  out  satisfactorily. 

Items  (86)  to  (88):  Number  of  Commutator  Bars. — Refer  Art. 
27,  page  93.  On  a  220- volt  machine,  the  potential  difference 
between  adjacent  commutator  segments  might  be  anything 
between  2.5  and  10  volts  (page  94).  If  we  provide  the  same 
number  of  commutator  bars  as  there  are  slots  on  the  armature, 
the  average  voltage  between  the  segments  at  full  load  would 

230  X  4 

be  about  — ?= —  =  8.63,  which  is  within  the  limits  obtained  in 
o/ 

practice.  At  the  same  time  commutation  will  be  very  much 
improved  by  having  a  smaller  number  of  turns  between  the 
tappings,  and  since  the  number  of  inductors  in  each  slot  is 
not  divisible  by  4,  we  shall  have  to  provide  57  X  3  =  171 
commutator  segments. 

Items  (93)  to  (99):  Dimensions  of  Brushes. — Unless  a  very 
soft  quality  of  carbon  is  used,  the  current  density  over  brush- 
contact  surface  does  not  usually  exceed  40  amp.  per  square 
inch.  Taking  35  as  a  suitable  value,  the  contact  surface  of  all 

S26 
brushes  of  the  same  sign  will  be  -^~-  =  9.32  sq.  in.,  or  4.66  sq. 

in.  per  brush  set.  If  the  brush  covers  three  bars,  the  width 
will  be  W  =  0.247  X  3  =  0.741  in.  Let  us  make  this  dimen- 
sion %  in.  The  total  length  of  brushes  per  set,  measured  in 
a  direction  parallel  to  the  axis  of  the  machine,  will  then  be 
lc  —  4.66^0.75  =  6.23,  or  (say)  6  in.,  which  can  be  made  up  of 
six  brushes  each  1  in.  by  %  in. 

"Item  (100) :  Length  of  Commutator. — The  spaces  between  brushes 
(which  will  depend  upon  the  type  of  brush  holder)  and  the 


224          PRINCIPLES  OF  ELECTRICAL  DESIGN 

clearances  at  the  ends,  including  an  allowance  for  "  staggering" 
the  brushes,  will  probably  require  a  minimum  axial  length  of 
commutator  surface  of  7J^  in. 

Item  (101):  Flux  Cut  by  End  Connections.—  Refer  Art.  48. 
Assuming  for  the  constant  in  formula  (72)  on  page  159,  the 
average  value  k  =  2.4,  we  have: 

$e  =  0.4  \/2  X  2.4  X  3  X  83.4  X  ^    X  2.75  [  (log.  y)  -  1  ] 
=  31,300  maxwells. 

Item  (102)  :  Slot  Flux.—  Refer  Art.  49.  The  equivalent  slot 
flux,  by  formula  (80),  page  164,  is 

1.6  X  TT  X  1  X  3  X  83.4  X  11  X  2.54 


6X05 


maxwells- 


Items  (103)  and  (104)  :  Average  Flux  Density  in  Commutating 
Zone.—  Refer  Art.  49.  By  formula  (81)  page  164: 

3>c  =  (2  X  11,730)  +  31,300  =  54,760    maxwells.     The 
average  density  is,  therefore,  by  formula  (83)  : 

54,760 
B<  =  1.083  X  11  X  6.46  =  712  gaUSSeS' 

Items  (105)  and  (106)  :  Flux  Densities  at  Beginning  and  End  of 
Commutation.  —  The  value  of  item  (104)  is  the  density  of  the 
magnetic  field  at  the  middle  of  the  zone  of  commutation.  After 
the  current  in  the  short-circuited  coil  has  passed  through  zero 
value  (a  condition  attained  only  when  the  coil  as  a  whole  is 
moving  in  a  neutral  field),  the  field  should  increase  in  strength 
until,  at  the  end  of  commutation,  it  is  of  such  a  value  as  to 
develop  ICR  volts  in  the  short-circuited  coil.  The  resistance,  R, 
of  the  coil  of  one  turn  is  0.00132  ohm  (item  (42)),  and  the  e.m.f. 
to  be  developed  in  the  coil  at  the  beginning  and  end  of  commu- 
tation is  therefore  0.00132  X  83.4  =  0.11  volt,  or  0.055  volt  in 
each  coil-side.  The  flux  to  be  cut  by  each  coil-side  to  develop 
this  voltage  is: 

Maxwells  per  centimeter  __  volts  X  108  _ 
of  armature  periphery     "  rate  of  cutting,  in  centimeters  per 

second 

=  0.055  X  10°  X  3)070  X?2  X  2.54 
=  3,530 


PROCEDURE  IN  DESIGN  OF  D.C.  GENERATOR    225 


and  this  corresponds  to  a  density  of 


3  530 
^  2  54  =   126  gausses. 


The  ideal  flux  density  at  beginning  of  commutation  is  therefore 
712  —  126  =  586  gausses,  and  at  the  end  of  commutation,  712+ 
126  =  838  gausses. 

Item  (107)  :  Sparking  Limits  of  Flux  Density  in  Zone  of  Com- 
mutation. —  Refer  Art.  52.     The  time  of  commutation,  in  seconds, 


5000  - 


4000  - 


3000 


2000 


1000  - 


1000  - 


I  k  j  i  h  g 

FIG.  85. — Flux  distribution  in  zone  of  commutation. 

is  approximately  tc  =  ^r  =  o  130  v  12  =  0-00176  sec.;  and,  by 
formula  (88)  page  177; 

$  =  %  X  0.00176  X  108 
=  132,000  maxwells. 

The  variation  in  flux  density  which  is  permissible  when  carbon 
brushes  of  medium  hardness  are  used,  will  therefore  be 

132,000 
1.083  X  11  X  6.45  ==1^20  gausses. 

A  portion  of  the  full-load  flux  curve  C  of  Fig.  84  has  been  re- 
drawn in  Fig.  85  to  a  larger  scale,  and  the  flux  density  required 

15 


226          PRINCIPLES  OF  ELECTRICAL  DESIGN 

to  obtain  ideal  commutation,  together  with  the  permissible 
variation  from  this  value,  are  indicated  on  the  same  diagram. 
It  will  be  seen  that  there  should  be  no  difficulty  in  obtaining 
sparkless  commutation  at  full  load  with  the  brushes  in  the  selected 
position  (over  the  point  k) ;  but  the  brush  might  be  moved  with 
advantage  4°  or  5°  nearer  to  the  leading  pole  tip;  and,  al- 
though the  final  flux  curve  C  would  be  slightly  modified,  it 
would  not  depart  materially  from  the  line  drawn  in  Fig.  85. 
The  angular  degrees  referred  to  are  so-called  " electrical"  de- 
grees, because  the  pole  pitch  has  been  divided  into  180  parts;  the 
displacement  referred  to  therefore  corresponds,  in  a  four-pole 
machine,  to  a  movement  of  2  to  2J£  actual  space  degrees. 

In  connection  with  Fig.  85,  it  should  be  observed  that  it  is 
only  in  the  case  of  a  full-pitch  winding  that  both  coil-sides  will 
be  moving  through  a  field  of  the  same  density  at  the  same 
instant  of  time.  In  the  design  under  consideration  the  pole 
pitch  is  equal  to  11J4  times  the  slot  pitch,  while  the  two  sides 
of  the  coil  would  probably  be  spaced  exactly  11  slot  pitches 
apart.  This  is  very  little  short  of  a  full-pitch  winding,  and  the 
flux  cut  by  the  two  sides  of  the  coil  is  very  nearly  the  same 
at  any  given  instant;  but  the  method  illustrated  by  Fig.  85  can, 
of  course,  be  used  for  determining  the  proper  brush  position  with 
short-pitch  as  well  as  with  full-pitch  windings. 

Item  (108) :  Brush  Pressure. — Refer  Arts.  53  and  54.  Assume 
Ij-^  lb.  per  square  inch. 

Items  (109)  to  (112):  Brush  Resistance  and  Losses. — Refer 
Arts.  53  and  54.  From  Fig.  68  (page  179)  we  find  the 
surface  resistance  of  hard  carbon  brushes  to  be  about  0.025  ohms 
per  square  inch  fora  current  density  of  36.2  (item  (93)).  The 
area  of  all  brushes  of  the  same  sign  is  9  sq.  in.  (item  (94)) ;  and  the 
total  brush  resistance  is  therefore 

°-°295  X  2  =  0.00556  ohm. 

The  calculated  IR  drop  is  0.00556  X  326  =  1.81  volts,  which 
should  be  increased  by  about  25  per  cent,  as  suggested  on  page 
181,  making  this  item  2.25  volts.  The  PR  loss  is  326  X  2.25  = 
730  watts.  In  these,  and  some  previous,  calculations,  the  value 
of  the  line  current  (item  (12))  has  been  used  in  place  of  the  total 
current  passing  through  the  armature  windings.  It  is  true 
that  an  allowance  should  have  been  made  for  the  shunt  exciting 


PROCEDURE  IN  DESIGN  OF  D.C.  GENERATOR    227 

current;  but  this  is  usually  an  unnecessary  refinement;  and  when 
calculating  brush  losses,  great  accuracy  in  results  is  not  attainable 
because  the  resistance  at  the  brush-contact  surface  is  always  a 
quantity  of  rather  doubtful  value. 

Item  (113):  Brush-friction  Loss.  —  Refer  Art.  54.  Assuming  a 
coefficient  of  friction  of  0.25,  the  brush-friction  loss  by  formula 
(89)  is 

0.25  X  1.5  X  18  X  600  X  13.5  X  TT  X  746 

Wf  -  "i2X3pOO~  =  324  WattS' 

Items  (115)  and  (116):  Cooling  Surface  and  Temperature  Rise 
of  Commutator.—  Refer  Art.  54  and  Fig.  70,  page  183.  The  radial 
height  of  the  risers  will  be  about  2  in.,  making  Dr  =  17%  in. 
The  radial  depth  of  the  exposed  ends  of  the  copper  bars  might 
amount  to  %  in.,  making  De  =  12  in.;  and  the  total  cooling 
surface  considered,  worked  out  as  explained  on  page  182,  amounts 
to  492  sq.  in. 

The  radiating  coefficient,  as  given  in  formula  (90),  is 

2  130 
°-°25  +  -  °'0463' 


whence    T  =  400  y  n  Vufis  =  46.3°C.,    which    is    permissible. 


Had  this  calculated  temperature  rise  exceeded  50°,  it  might 
have  been  necessary  to  increase  the  axial  length  of  the  com- 
mutator, or  reduce  the  losses  by  using  a  soft  quality  of  carbon 
brush  and  perhaps  a  lighter  pressure  at  the  contact  surface. 
In  some  cases  special  ventilating  ducts  are  provided  inside  the 
commutator;  but  these  should  not  be  necessary  in  a  machine  of 
the  type  and  size  considered. 

Item  (117):  Leakage  Coefficient.  —  Refer  Art.  56,  page  186. 
The  value  of  this  item  was  estimated  at  1.2  in  connection  with 
the  shaping  of  the  pole  shoe  (item  (68)). 

Items  (118)  to  (121):  Flux  Density  in  Pole  Core.—  A  cylindrical 
pole  core  10  in.  in  diameter  has  been  decided  upon  (item  (68)). 
The  area  of  cross-section  is  therefore  78.54  sq.  in.,  and  the 
full-load  flux  density  in  the  pole  core  near  the  yoke  ring  is 

6,970,000  X  1.2 
78.54  X  6.45      =  16'50°  gausses* 

Item  (122):  Radial  Length  of  Pole.—  Refer  Art.  55,  page  186. 
The  full-load  ampere-turns  per  pole  for  air  gap,  teeth  and  slots 
amount  to  about  6,000  (see  Fig.  83).  Then,  by  formula  (91), 


228 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


the  length  of  winding    space    should    be    c  = 


6,000 

875 


6.85  in. 


Let  us  make  the  cylindrical  pole  core  7  in.  long,  which  will  de- 
termine the  inside  diameter  of  the  yoke  ring.  This  will  have 
to  be  about  38  in.  as  shown  in  Fig.  86. 

Items  (123)  to  (127):  Dimensions  of  Yoke  Ring. — Assuming  a 
density  of  15,000  gausses  in  the  cast-steel  yoke  ring,  the  cross- 
section  will  be 

6,970,000  X  1.2 
2  X  15,000  X  6.45  " 

The  dimensions  can  now  be  determined,  and  the  lengths  of  the 
flux  paths  obtained  from  Fig.  86. 


FIG.  86. — Magnetic  circuit  of  four-pole  dynamo. 

Item  (128):  Open-circuit  Saturation  Curve. — Refer  Chap.  IX, 
Arts.  55,  56,  and  57.  Also  Art.  16  of  Chap.  III.  The  calcula- 
tions of  the  total  ampere-turns  on  each  pole  of  the  machine,  to 
develop  on  open  circuit  a  given  voltage,  are  shown  in  the  accom- 
panying table.  Suitable  values  of  terminal  voltage  are  selected 
to  obtain  points  on  the  saturation  curve.  One  of  the  values 
should  be  slightly  higher  than  the  developed  e.m.f.  under  full- 
load  conditions.  It  is  not  necessary  to  make  the  calculations 
for  very  low  voltages  because  the  reluctance  of  the  iron  parts 
of  the  magnetic  circuit  is  then  negligible.  For  each  selected 
value  of  the  developed  e.m.f.,  the  ampere-turns  for  the  com- 
plete magnetic  circuit  are  calculated  exactly  as  explained  in 


PROCEDURE  IN  DESIGN  OF  D.C.  GENERATOR    229 


Chap.  Ill,  Art.  16,  in  connection  with  the  horseshoe  lifting  mag- 
net. The  useful  flux  entering  the  armature  must  be  multiplied 
by  the  leakage  factor  to  obtain  the  total  flux  in  the  yoke  ring; 
and,  in  the  case  of  the  pole  cores,  the  approximation  suggested 
in  Art.  16  (page  60)  may  be  used.  The  average  value  of  the 
pole-core  density,  for  use  in  calculating  the  ampere-turns  required, 


s 


In  this  instance  By  =  1.2BP',  whence  Bc  =  1.133BP,  the  mean- 
ing of  which  is  that  the  density  in  the  pole  core  is  calculated 
on  the  assumption  that  the  leakage  factor  is  1.133  instead  of 
1.2,  as  used  for  estimating  the  flux  in  the  frame. 

OPEN-CIRCUIT  SATURATION 
(Table  for  Calculating  Ampere-turns  per  Pole  for  Total  Magnetic  Circuit) 


No-load  voltage  

Flux     entering     armature     per     pole 

(maxwells)  

245 
7,160,000 

230 

6,725,000 

210 

6,140,000 

190 
5,560,000 

Flux  density  (lines 
per  square  inch) 

Armature  core  (36  sq.  in.)  
Air  gap  (maximum  value)  
Pole  core  (78.54  sq.  in.)  
Yoke  ring  (43.2  sq.  in.)  

99,400 
56,100 
103,400 
99,400 

93,400 
52,600 
97,100 
93,400 

85,250 
48,000 
88,700 
85,250 

77,250 
43,400 
80,300 
77,250 

a 

Armature  

80 

42 

20 

11 

!l 

Pole  core  

104 

64 

26 

14 

<D   "- 

Yoke  

80 

42 

20 

11 

i 

Armature  (o  =  5.1)  

408 

214 

102 

56 

l« 

Pole  core  (c  —  8) 

832 

512 

208 

112 

g| 

3«* 

Yoke  (y  =  15) 

1,200 

630 

300 

165 

0 

Air  gap  and  teeth 

5,750 

5,300 

4,800 

4  250 

Total  ampere-turns  

8,190 

6,656 

5,410 

4,583 

230 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


The  ampere-turns  per  inch  are  read  off  the  upper  curve  of 
Fig.  3  (page  17),  and  the  lengths  of  the  various  parts  of  the 
magnetic  circuit  are  taken  from  Figo  86.  The  length  of  the 
iron  path  in  the  armature  core  is  taken  as  one-third  of  the  pole 
pitch,  as  suggested  in  Art.  57.  The  ampere-turns  for  the  air 
gap,  teeth  and  slots  are  read  directly  off  the  curve  a,  b,  c,  dt 
e,  /,  of  Fig.  82  (page  219),  for  the  density  corresponding  to  the 
maximum  value  of  the  flux  curve  A  of.  Fig.  84.  It  is  assumed  that 


wo 

240 
230 
220 
210 
200 
190 
«180 
>170 
160 
150 
140 
130 
120 

^ 

^ 

•** 

G 

/ 

/" 

F 

^ 

/f 

/, 

' 

/ 

/ 
/ 
/ 

/ 

j 

/ 

/ 
/ 

/ 

/ 
/ 

/ 

/ 

/ 
/ 

/ 
/ 

/ 

/ 

Sh 

unt. 

Open 

Oirc 

iit^ 

-> 

*ti 

)isto 

tion 

S 

mnt. 

Full 

Loac 

Se: 

ies 

110 

3000 


4000 


8000 


9000 


5000          6000  7000 

Ampere  -Turns  per  Pole 

FIG.  87.  —  Open-circuit  saturation  curve  (numerical  example). 

the  shape  of  this  curve  remains  unaltered,  and  that  the  maximum 
ordinate  is  directly  proportional  to  the  developed  voltage. 

The  curve,  Fig.  87,  is  plotted  from  the  results  of  these  calcu- 
lations. It  shows  the  connection  between  developed  e.m.f.  (or 
open-circuit  terminal  voltage)  and  the  corresponding  ampere- 
turns  of  excitation  per  pole.  The  shunt  ampere-turns  on  open 

230 
circuit  are  5,930,  and  at  full  load,  5,930  X  220  =  6,200.     The 

ampere-turns  in  the  series  winding  are  7,850  —  6,200  =  1,650, 
which  includes  the  ampere-turns  to  compensate  for  armature 


PROCEDURE  IN  DESIGN  OF  D.C.  GENERATOR     231 

demagnetization  and  distortion.  This  correction,  amounting  to 
400  ampere-turns,  is  obtained  from  Fig.  82,  where  this  number 
of  ampere-turns  is  seen  to  be  necessary  to  raise  the  flux  density 
under  the  center  of  the  pole  from  7,260  to  7,800  gausses,  as 
explained  on  page  222. 

Items  (133)  to  (137)  :  Shunt  Field  Winding.—  Refer  Art.  58, 
Chap.  IX,  and  Art.  10,  Chap.  II.  The  length  of  winding  space 
for  the  shunt  may  be  determined  by  dividing  the  total  length 
available  for  the  windings  in  the  proportion  of  the  ampere- 
turns  in  shunt  and  series  coils  respectively.  The  length  of  the 

6  200 
cylindrical  core  is  7  in.,  and  7  X    ^    =  5,53.     Some  allowance 


should  be  made  for  external  insulation,  and  the  net  length  of 
winding  space  for  the  shunt  coils  might  be,  say,  5  in.  Let  us 
assume  the  total  thickness  of  winding  to  be  2  in.  The  inside 
diameter  of  the  winding  might  be  10)^  in.,  making  the  average 
diameter  1234  in.,  and  the  mean  length  per  turn,  38.5  in.  We 
shall  suppose  that  the  shunt  rheostat  absorbs  15  per  cent,  of 
the  voltage  on  open  circuit;  which  leaves  187  volts  across  the 
s,hunt  winding.  By  formula  (26)  on  page  42,  we  have: 

38.5  X  5,930  X  4 
(ro)  =  ~       —      -          =  4>880 


Referring  to  the  wire  table  on  page  34,  the  standard  size  of 
wire  of  cross-section  nearest  to  this  calculated  value  is  No. 
13  B.  &  S.  gage.  This  can  be  used  if  the  rheostat  is  arranged  to 
reduce  the  voltage  across  the  winding  in  the  proper  proportion. 
The  number  of  turns  per  inch  is  11.8,  from  which  it  is  seen  that 
1,360  turns  can  be  wound  in  the  space  available. 
The  resistance  of  all  the  four  coils  in  series  is 

38.5  X  1,360  X  4  X  2.328 

12  X  1,000  l0'5  °hmS)  at 

The  current,  under  open-circuit  conditions,  is  .,'     ^  =  4.36 

230 
amp.,  and  at  full  load  (item  (136))  it  is  4.36  X  220=  4.56   amp. 

This  is  only  1.4  per  cent,  of  the  line  current;  a  low  value,  which 
might  perhaps  be  increased  in  order  to  reduce  the  amount  of 
copper  in  the  field  coils  if  the  temperature  rise  is  not  excessive. 
Items  (139)  to  (142):  Series  Field  Coils.—  Refer  Art.  58,  page 
192.  The  series  turns  may  be  placed  at  either  end  of  the  pole, 


232          PRINCIPLES  OF  ELECTRICAL  DESIGN 

preferably  near  the  pole  shoe.  The  space  available  in  a  radial 
direction  is  about  7  —  5J£  =  1%  in.  The  number  of  turns 

per  pole  is    '        =  5.06.    Let  us  put  5^  turns  on  each  pole,  and 

make  the  final  adjustment  by  means  of  a  diverter.     The  current 

1  650 

through  the  series  winding  will  therefore  be    '  ,     =  300  amp. 

o.o 

The  total  depth  of  winding  might  be  about  the  same  as  for  the 
shunt  coils,  i.e.,  2  in.  The  mean  length  per  turn  would  then  be 
38.5  in.,  and  the  total  length,  4  X  5.5  X  38.5  =  847  or,  with  an 
allowance  for  connections,  say,  890  in.  Assuming  a  current 
density  of  1,200  amp.  per  square  inch,  the  cross-section  would  be 

OQA 

^  2QQ  =  0.25  sq.  in.     This  winding  may  consist  of  flat  copper 

strip  wound  on  edge,  or  of  any  other  shape  of  conductor  of  this 
cross-section.  If  preferred,  two  or  more  conductors  of  some  stock 
size  can  be  connected  in  parallel  to  make  up  a  total  cross-section 
of  about  0.25  sq.  in.  The  space  available  is  more  than  sufficient, 
and  we  shall  assume  for  the  present  that  the  cross-section  is 
exactly  0.25  sq.  in.,  or  318,000  circular  mils.  The  resistance,  at 

890 
60°C.,  will  then  be  Q1Qnnn  =  0.0028  ohm.     The  drop  in  volts  in 


the  series  winding  is  therefore  0.0028  X  300  =  0.84,  which,  being 
very  small,  may  be  increased  if  it  is  found  that  the  temperature 
rise  is  appreciably  below  the  specified  limit. 

Items  (143)  and  (144)  :  Temperature  Rise  of  Field  Coils.  —  Refer 
Art.  59,  Chap.  IX.  The  area  of  the  two  cylindrical  surfaces  is 
approximately  7  X  7r(10  -f  14)  =  528  sq.  in.  The  area  of  the 

two  ends  is  2  X  j  (14    -  To  )  =  151  sq.  in.     The  total  cooling 

surface  of  all  the  field  windings  is  therefore  679  X  4  =  2,720  sq. 
in.,  approximately. 

The  PR  loss  at  full  load  in  the  shunt  winding  is  (4.56)  2  X  38.4 
=  800  watts;  and  the  PR  loss  in  the  series  coils  is  (300)  2  X 
0.0028  =  252  watts,  making  a  total  loss  of  1,052  watts. 

The  cooling  coefficient,  as  given  by  curve  A  of  Fig.  76  (page 
194),  is  0.009,  and  the  temperature  rise  will  therefore  be 

1,052 
0.009  X  2,720 

This  is  a  little  higher  than  the  specified  limit  of  40°,  and  if  the 
cooling  coefficient  could  be  relied  upon  for  the  accurate  prede- 


PROCEDURE  IN  DESIGN  OF  D.C.  GENERATOR    233 

termination  of  the  temperature  rise,  it  would  be  necessary  either 
to  increase  the  weight  of  copper  in  the  coils,  or  to  sectionalize 
the  windings  so  as  to  improve  the  ventilation.  The  latter  course 
would  be  the  right  one  in  this  case  since  the  copper  loss  is  not  by 
any  means  excessive,  and  it  would  be  desirable  to  reduce  rather 
than  increase  the  amount  of  copper  in  the  field  windings. 

Items  (145)  and  (146):  Resistance  of  Diverter. — Assuming  the 
"long  shunt"  connection,  the  series  current  passing  through  the 
diverter  will  be  30.56  amp.,  and  the  resistance  of  the  diverter 
must  therefore  be 

son 

0.0028  X  qTTKA  =  °-0275  ohm- 

OO.OD 

A  resistance  slightly  greater  than  this  should  be  provided,  of  a 
material  and  cross-section  capable  of  carrying  at  least  40  amp. 
without  undue  heating.  The  final  adjustment  can  then  be  made 
when  the  machine  is  on  the  test  floor. 

Item  (147):  New  Calculation  of  Losses  in  Teeth. — Refer  Art.  60, 
Chap.  IX.  The  maximum  value  of  the  air-gap  density  under 
full-load  conditions  may  be  read  off  curve  C  of  Fig.  84  (page  221) 
where  it  is  seen  to  be  10,800  gausses.  The  corresponding  tooth 
density,  as  read  off  Fig.  81  (page  216),  is  23,100.  This  is  the 
density  at  the  narrowest  part  of  the  tooth.  On  the  assumption 
that  flux  neither  enters  nor  leaves  the  tooth  up  to  a  distance  de 
from  the  bottom  of  the  slot  (see  Fig.  38,  page  122),  the  tooth 
densities  at  the  three  sections  considered  are  Bw  =  18,870,  Bm  = 
21,000,  and  Bn  =  23,100  gausses.  Referring  to  Fig.  34  (page 
102),  we  find  the  watts  per  pound  corresponding  to  these  den- 
sities to  be  4.1,  4.8,  and  5.5,  respectively,  the  mean  value  being 
4.8.  The  total  weight  of  iron  in  the  teeth  (item  (57))  is  75  lb.,  and 
the  corrected  total  loss  in  the  teeth  is  75  X  4.8  =  360  watts. 

Items  (148)  and  (149):  Efficiency  at  Any  Output.— Refer  Art. 
60,  Chap.  IX.  The  efficiency  table  on  page  235  requires  but 
little  explanation.  Each  column  stands  for  a  particular  output, 
expressed  as  a  fraction  of  rated  full  load.  The  terminal  voltage 
is  calculated  on  the  assumption  that  it  conforms  to  a  straight- 
line  law  based  on  the  known  full-load  and  no-load  voltages. 

The  windage  and  friction  loss  is  taken  as  1.8  per  cent,  of  the 
full-load  output  (see  page  196). 

The  core  loss — which  includes  the  corrected  tooth  loss — is  the 
calculated  full-load  value.  It  will  actually  vary  somewhat  with 


234 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


the  load  (and  developed  voltage),  but  for  practical  purposes  may 
be  assumed  constant. 

The  constant  losses  are  made  up  as  follows: 

Windage  and  bearing  friction  =  0.018  X  75,000  =  1,350  watts. 

Brush  friction  (item  (113)) =     324  watts. 

Iron  loss  in  core  and  teeth  (item  60) =  1,644  watts. 


Total =  3,318  watts. 

The  full-load  current  in  the  armature  is  the  line  current  plus 
the  shunt  current  (item  (136)),  and  since  the  armature  resistance 


i.o 


g.7 


.6 


20  40  60  80  100 

Output  _  ( Percentage  of  Full  Load ) 

FIG.  88. — Efficiency  curve. 


120 


140 


is  known  (item  (44)),  the  PR  loss  in  the  armature  at  different  loads 
can  readily  be  calculated. 

The  brush-contact  PR  loss  is  obtained  by  referring  to  Fig.  69 
(page  179)  and  adding  25  per  cent,  to  the  calculated  losses. 

We  shall  assume  the  "long  shunt"  connection,  which  means 
that  the  series  winding  and  diverter  will,  together,  carry  the  full 
armature  current,  and  the  series  field  PR  losses  will  therefore  be 
directly  proportional  to  the  copper  losses  in  the  armature. 

Fig.  88  is  the  efficiency  curve  plotted  from  the  figures  in  the 
table.  The  full-load  efficiency  is  0.915,  and  judging  by  the  shape 


PROCEDURE  IN  DESIGN  OF  D.C.  GENERATOR     235 


of  the  curve,  the  maximum  efficiency  of  0.92  will  probably  be 
obtained  with  an  overload  of  about  25  per  cent. 

EFFICIENCY  TABLE 


Load  output  

0 

M 

H 

M 

1 

IK 

Terminal  voltage  

220 

222.5 

225 

227.5 

230 

232.5 

Line  current  

0 

84.3 

167 

247 

326 

403 

Constant  power  loss.  .  

3,318 

3,318 

3,318 

3,318 

3,318 

3,318 

Armature  I2R  loss  

111 

419 

897 

1,540 

2,342 

Brush-contact  I*R  loss  

152 

322 

528 

744 

964 

Series  field  and  diverter  

20 

76 

162 

278 

423 

Shunt  field  and  rheostat  

960 

980 

1,000 

1,025 

1,050 

1,070 

Total  loss  

4,278 

4,582 

5,135 

5,930 

6,930 

8,117 

Output  (watts)  

0 

18,750 

37,500 

56,250 

75,000 

93,750 

Input  (watts)  

4,278 

23,332 

42,635 

62,180 

81,930 

101,867 

Efficiency  (per  cent.)  

0 

80.3 

87.8 

90.5 

91.5 

92 

Referring  to  the  usual  efficiencies  of  commercial  machines  as 
given  on  page  197,  it  is  seen  that  the  calculated  value  of  91.5 
compares  favorably  with  the  average  value  of  91.2  for  a  75-kw. 
dynamo. 

64.  Design  of  Continuous-current  Motors. — The  dynamo 
being  a  reversible  machine,  may  be  used  as  a  generator  to 
convert  mechanical  into  electrical  energy,  or  as  a  motor  to  convert 
electrical  into  mechanical  energy.  If  the  machine  is  to  be  used 
as  a  motor,  the  efficiency  should  first  be  estimated  by  referring 
to  the  figures  on  page  197.  This  efficiency,  in  the  case  of  a 

.    ,,         ,.    output 
motor,  is  the  ratio  - — £— -,  whence 


Kw.  = 


horsepower  X  746 


efficiency  X  1,000 

and  the  design  may  be  proceeded  with  exactly  as  if  the  machine 
were  to  be  used  as  a  generator  to  give  this  particular  kilowatt 
output  at  the  specified  speed. 

It  is  even  more  important  in  a  motor  than  in  a  generator 
that  the  machine  should  work  sparklessly  at  all  loads  without 
change  of  the  brush  position.  The  specification  usually  calls  for 
operation  without  destructive  sparking  from  zero  load  to  25 
per  cent,  overload,  with  the  brushes  in  a  fixed  position.  If  the 
direction  of  revolution  of  the  motor  is  to  be  reversible,  it  is  neces- 
sary for  the  brushes  to  be  on  the  geometric  neutral  line,  a  con- 
dition which  is  usually  met  by  providing  commutating  interpoles. 


236          PRINCIPLES  OF  ELECTRICAL  DESIGN 

On  account  of  the  conditions  under  which  they  have  to  operate, 
dynamo  machines  when  used  as  motors  are  more  often  totally 
enclosed  than  when  used  as  generators.  In  the  case  of  the  larger 
units  forced  ventilation  would  then  be  resorted  to,  but  the 
smaller  sizes  may  be  self-cooling.  The  temperature  rise  is  then 
largely  equalized  throughout  the  machine,  and  somewhat  higher 
surface  temperatures  are  allowable  than  in  the  case  of  open-type 
machines.  A  temperature  rise  of  60°,  by  thermometer,  is  allow- 
able inside  the  machine  but  this  means  that  the  temperature 
rise  of  the  enclosing  case  must  be  considerably  less  than  this,  say 
35°  or  40°C. 

In  the  absence  of  data  on  the  particular  type  of  enclosed  motor 
under  consideration,  a  cooling  coefficient  of  0.008  to  0.01  may 
be  used.  This  figure  denotes  the  number  of  watts  that  can  be 
radiated  per  degree  Centigrade  rise  of  temperature  from  every 
square  inch  of  the  entire  external  surface  of  the  enclosed  motor. 


CHAPTER  XI 

DESIGN  OF  ALTERNATORS— FUNDAMENTAL 
CONSIDERATIONS 

65.  Introductory. — In  the  continuous-current  dynamo  the  func- 
tion of  the  commutator  is  merely  to  rectify  the  armature  cur- 
rents in  order  that  a  machine  with  alternating  e.m.fs.  generated 
in  its  windings  shall  deliver  unidirectional  currents  at  the  ter- 
minals. It  may  therefore  be  argued  that  the  design  of  alter- 
nating-current generators  should  be  taken  up  before  that  of  D.C. 
dynamos,  the  changes  caused  by  the  addition  of  the  commutator 
being  considered  in  the  second  place.  There  are,  however,  many 
matters  of  importance  to  be  considered  in  connection  with  an 
alternating-current  generator,  which  have  no  part  in  the  design  of 
a  continuous-current  dynamo.  Among  these  may  be  mentioned 
the  effects  due  to  changes  in  wave  shapes  of  e.m.f.  and  current;  the 
importance  of  the  inductance,  not  only  of  the  armature  itself,  but 
also  of  the  circuit  external  to  the  generator;  and  the  fact  that  the 
voltage  regulation  depends  not  only  on  the  IR  drop,  but  also  on 
the  power  factor  of  the  load,  i.e.,  on  the  phase  displacement  of  the 
current  relatively  to  the  e.m.f.  The  problems  to  be  solved  being 
somewhat  less  simple  than  those  connected  with  continuous-current 
machines,  the  writer  believes  that  the  arrangement  of  the  subject 
as  followed  in  this  book  is  justified. 

It  is  proposed  to  treat  the  design  of  A.C.  machines  as  nearly 
as  possible  on  the  lines  followed  in  the  D.C.  designs.  In  order 
to  avoid  unnecessary  repetitions,  references  will  be  made  to  pre- 
vious chapters  and  stress  will  be  laid  on  the  essentials  only, 
particular  attention  being  paid  to  the  points  of  difference  between 
A.C.  and  D.C.  machinery. 

The  design  of  asynchronous  generators  will  not  be  touched 
upon.  This  type  of  machine  is  essentially  an  induction  motor 
reversed,  the  rotor,  with  its  short-circuited  windings,  being  me- 
chanically driven.  The  writer  has  explained  elsewhere  the  prin- 
ciples underlying  the  working  of  these  machines,1  and  since  they 

1  ALFRED  STILL:  "Polyphase  Currents,"  WHITTAKER  &  Co. 

237 


238          PRINCIPLES  OF  ELECTRICAL  DESIGN 

are  of  a  type  not  commonly  met  with,  they  will  not  again  be 
referred  to. 

The  remainder  of  this  book  will  be  devoted  to  a  study  of  the 
synchronous  alternating-current  generator,  and  since  multipolar 
polyphase  generators  with  stationary  armatures  are  more  com- 
mon than  any  other  type,  they  will  receive  more  attention  than 
the  less  frequently  seen  designs;  but  the  case  of  the  high-speed, 
steam-turbine-driven  units,  with  a  small  number  of  poles  and 
distributed  field  windings,  will  also  be  considered. 

Apart  from  the  absence  of  commutator,  the  chief  point  of  dif- 
ference between  an  A.C.  and  D.C.  generator  is  that  the  frequency 
of  the  former  is  specified,  whereas,  in  the  latter,  this  is  a  matter 
which  concerns  the  manufacturer  only.  It  follows  that,  for  a 
given  speed,  the  number  of  poles  is  determined  by  the  frequency 
requirements,  and  this  fact  necessarily  influences  the  design.  In 
Europe  a  frequency  of  50  cycles  per  second  is  common,  the  idea 
being  that  this  is  high  enough  for  lighting  purposes  while  being 
sufficiently  low  to  allow  of  the  same  circuits  being  used  occasion- 
ally for  power  purposes  also.  A  lower  frequency  is  usually  to  be 
preferred  for  power  schemes,  and  the  standards  in  America  are 
25  cycles  for  power  purposes  and  60  cycles  for  lighting. 

66.  Classification  of  Synchronous  Generators. — It  is  well  to 
distinguish  between  two  classes  of  alternators: 

1.  Machines  with  salient  poles,  driven  at  moderate  speeds  by 
belt,   or  direct-connected    to  reciprocating    steam,   gas,   or  oil 
engines,  or  to  water  turbines.     The  peripheral  speed  of  the  ro- 
tating part  (usually  the  field  magnets)  will  generally  lie  between 
the  limits  of  3,000  and  8,000  ft.  per  minute. 

2.  Machines  direct-coupled  to  high-speed  steam  turbines,  in 
which  the  peripheral  velocity  usually  exceeds  12,000  ft.  per  min- 
ute, is  commonly  about  18,000,  and  may  attain  24,000  ft.  per  min- 
ute.    In  these  machines  the  field  system  is  always  the  part  that 
rotates;  the  number  of  poles  is  small,  and  although  salient  poles 
are  sometimes  used  on  the  lower  speeds,  the  cylindrical  field 
magnet  with  distributed  windings  is  more  common.     The  me- 
chanical problems  encountered  in  the  design  of  these  high-speed 
machines  are  relatively  of  greater  importance  than  the  electrical 
problems;  but  since  these  are  beyond  the  scope  of  this  book, 
they  will  not  be  considered  in  detail.     Such  differences  as  occur 
in  the  electrical  calculations  will  be  pointed  out  as  the  work 
proceeds. 


DESIGN  OF  ALTERNATORS  239 

Referring  again  to  the  type  of  generator  to  operate  at  moderate 
speeds,  it  is  not -necessary  for  the  field  to  rotate,  and  small  units, 
especially  when  the  voltage  is  low,  may  be  built  generally  on  the 
same  lines  as  D.C.  dynamos,  i.e.,  with  rotating  armatures  and 
an  external  crown  of  poles.  In  this  case  the  commutator  is  re- 
placed by  two  or  more  slip  rings  connected  to  the  proper  points 
on  the  armature  winding.  For  a  three-phase  generator  three 
slip  rings  are  required,  and  since  two  rings  only  are  necessary  if 
the  field  rotates,  the  design  with  stationary  armature  is  the  more 
common.  It  should  also  be  observed  that  the  insulation  of  the 
slip  rings  for  the  comparatively  low  voltage  of  the  exciting  circuit 
offers  no  difficulty,  whereas  the  insulation  of  the  alternating- 
current  circuit  may  have  to  withstand  fairly  high  pressures. 

The  field  magnet  windings  of  salient  pole  alternators  are  gen- 
erally similar  to  those  of  D.C.  dynamos,  that  is  to  say,  all  the 
poles  are  provided  with  exciting  coils.  Machines  have  been 
built  in  the  past  with  windings  on  alternate  poles  only;  with  a 
single  exciting  coil  (as  in  the  "MORDEY"  flat-coil  alternator); 
and,  again,  without  any  windings  on  the  rotating  part.  The 
latter  type  is  known  as  the  inductor  alternator,  and  the  field 
winding  is  then  put  on  the  stationary  armature  rings,  thus  dis- 
pensing with  slip  rings  for  the  collection  of  either  alternating  or 
continuous  current.  Iron  projections  on  the  rotating  part  so 
modify  the  reluctance  of  the  magnetic  circuit  through  the  arma- 
ture coils  that  alternating  e.m.fs.  are  generated  therein;  but  these 
machines,  together  with  those  having  single  exciting  coils,  have 
the  disadvantage  that  the  magnetic  leakage  is  very  great  and 
the  design  therefore  uneconomical. 

67.  Number  of  Phases. — Whether  a  machine  is  to  supply 
single-phase,  two-phase,  or  three-phase,  currents  does  not  ap- 
preciably affect  the  design.  The  calculations  on  a  machine  for 
a  large  number  of  phases  are  not  more  difficult  than  when  the 
number  of  phases  is  small.  The  theory  of  the  single-phase  gen- 
erator is,  in  fact,  somewhat  less  simple  than  that  of  the  polyphase 
machine.  In  the  succeeding  articles  it  is  the  three-phase  gen- 
erator that  we  shall  mainly  have  in  mind,  because  it  is  the  most 
commonly  met  with,  but  the  points  of  difference  in  the  electrical 
design  of  single-phase  and  polyphase  generators  will  be  pointed 
out  as  they  arise. 

Whatever  may  be  the  type  of  machine,  or  number  of  poles, 
we  may  consider  the  armature  conductors  to  be  cut  by  the  mag- 


240 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


netic  lines  in  the  manner  indicated  in  Fig.  89.  Here  we  have  a 
diagrammatic  representation  of  single-phase,  two-phase,  and 
three-phase,  windings.  In  each  case,  the  system  of  alternate 
pole  pieces  is  supposed  to  move  across  the  armature  conductors 
in  the  direction  indicated  by  the  arrow.  It  will  be  noted  that 
the  conductors  of  each  phase  are  shown  connected  up  to  form  a 
simple  wave  winding;  but  this  is  only  done  to  simplify  the  dia- 


FIG.  89. — Single-phase,  two-phase,  and  three-phase,  armature  windings. 

gram,  and  it  will  be  readily  understood  that  each  coil  may  con- 
tain a  number  of  turns,  attention  being  paid  to  the  manner  of 
its  connection  to  the  succeeding  coil,  in  order  that  the  e.m.fs. 
generated  in  the  various  coils  shall  not  oppose  each  other. 

The  upper  diagram  shows  a  single  winding,  in  which  an  alter- 
nating e.m.f.  will  be  generated.  In  the  middle  diagram  there 
are  two  distinct  windings,  A  and  B,  so  placed  on  the  armature 
surface  that  the  complete  cycle  of  e.m.f.  variations  induced  in 


DESIGN  OF  ALTERNATORS  241 

A  will  also  be  induced  in  B,  but  after  an  interval  of  time  repre- 
senting a  quarter  of  a  period.  This  diagram  shows  the  positions 
of  the  poles  at  the  instant  when  the  e.m.f.  in  A  is  at  its  maximum, 
while  in  B  it  is  passing  through  zero  value.  From  these  two 
windings  we  can,  therefore,  obtain  two-phase  currents  with  a 
phase  displacement  of  90  electrical  degrees. 

In  the  bottom  diagram  the  arrangement  of  three  windings  is 
shown,  from  which  three-phase  currents  can  be  obtained,  with  a 
phase  angle  between  them  of  120  degrees,  or  one-third  of  a  cycle. 
It  will  be  seen  that,  at  the  instant  corresponding  to  the  relative 
positions  of  coils  and  poles  as  indicated  on  the  diagram,  the  e.m.f. 
in  A  is  at  its  maximum,  while  in  B  and  C  it  is  of  a  smaller  value 
and  in  the  opposite  direction. 

68.  Number  of  Poles.     Frequency. — For  a  given  frequency 
the  number  of  poles  will  necessarily  depend  upon  the  speed. 

Thus  p  =  -»T^I  where  N  stands  for  the  speed  in  revolutions  per 

minute.  Since  /  is  usually  either  25  or  60,  it  follows  that  N  must 
be  some  definite  multiple  of  the  number  of  poles  p. 

69.  Usual  Speeds  of  A.C.  Generators. — The  speed  at  which  a 
machine  of  a  given  kilowatt  output  should  be  driven  will  depend 
upon  the  prime  mover.     The  speed  may  be  very  low,  as  when 
the  generator  is  direct-coupled  to  a  slow-speed  steam  engine  or 
low-head  waterwheel.     Higher  speeds  are  obtained  when  the 
generator  is  belt-driven  or  direct-coupled  to  high-speed  steam  or 
oil  engines.     Very  high  speeds  are  necessary  when  the  generator 
is  direct-connected  to  a  steam  turbine. 

For  usual  speeds  the  reader  is  referred  to  the  table  on  page 
81,  the  values  there  given  being  applicable  to  both  D.C.  and 
A.C.  machines.  In  hydro-electric  work  the  generator  is  usually 
direct-coupled  to  the  waterwheel,  the  speed  of  which  will  be 
high  in  the  case  of  impulse  wheels  working  under  a  high  head. 
As  an  example,  a  PELTON  waterwheel  to  develop  1,500  hp.  under 
a  head  of  1,000  ft.  would  have  a  wheel  about  5  ft.  in  diameter, 
running  at  500  revolutions  per  minute.  This  would  be  suitable 
for  direct  coupling  to  a  six-pole  25-cycle  generator. 

In  the  case  of  steam  turbines — with  a  blade  velocity  of  about 
5  miles  per  minute — the  speeds  are  always  very  high.  The  actual 
speed  best  suited  to  a  given  size  of  unit  will  depend  upon  the 
make  of  the  turbine,  but  the  following  table  gives  the  approxi- 
mate range  of  speeds  covered  by  modern  steam  turbines. 

16 


242          PRINCIPLES  OF  ELECTRICAL  DESIGN 

Output,  kilowatts  Approximate  range  of  speed, 

revolutions  per  minute 

2,000 3,000  to  6,000 

5,000 2,000  to  4,000 

10,000 1,200  to  2,500 

20,000 900  to  1,800 

70.  E.m.f.  Developed  in  Windings. — 

Let  $  =  flux  per  pole  (maxwells). 
N  =  revolutions  per  minute. 
p  =  number  of  poles. 

The  flux  cut  per  revolution  is  then  ^  X  p  and  the  flux  cut  per 

N 
second  is  $p  «g.     The  average  value  of  the  e.m.f.  developed  in 

each  armature  conductor  must  therefore  be 

$pN 

Ec  (mean)  =  6Q  x  1Q8  volts. 

If  the  space  distribution  of  the  flux  density  over  the  pole  pitch 
follows  the  sine  law,  the  virtual  value  of  the  e.m.f.  is  1.11  times 
the  mean  value.  In  other  words,  the  form  factor,  or  ratio 

r.m.s.   value  .      TT 

=— .  is  0    ,-.  or  1.11.  in  the  case  of  a  sine  wave. 

mean   value       2v2 

Concentrated  and  Distributed  Windings. — If  each  coil-side  may 
be  thought  of  as  occupying  a  very  small  width  on  the  armature 
periphery,  and  if  the  coil-sides  of  each  phase  winding  are  spaced 
exactly  one  pole  pitch  apart,  the  arrangement  would  constitute 
what  is  usually  referred  to  as  a  concentrated  winding.  With  a 
winding  of  this  kind,  all  conductors  in  series  in  one  phase  would 
always  be  similarly  situated  relatively  to  the  center  lines  of  the 
poles,  and  the  curve  of  induced  e.m.f.  would  necessarily  be  of 
the  same  shape  as  the  curve  of  flux  distribution  over  the  armature 
surface.  In  practice,  a  winding  with  only  one  slot  per  pole  per 
phase  would  be  thought  of  as  a  concentrated  winding.  When 
there  are  two  or  more  slots  per  pole  per  phase,  the  winding  is 
said  to  be  distributed;  and  since  the  conductors  of  any  one 
phase  cover  an  appreciable  space  on  the  armature  periphery, 
all  the  wires  that  are  connected  in  series  will  not  be  moving  in 
a  field  of  the  same  density  at  the  same  instant  of  time.  Except 
in  the  case  of  a  sine-wave  flux  distribution,  the  form  factor  may 
depend  largely  upon  whether  the  winding  is  concentrated  or 
distributed.  The  wave  shape  of  the  developed  voltage  can 
always  be  determined  when  the  flux  distribution  is  known;  but, 


DESIGN  OF  ALTERNATORS  243 

in  the  preliminary  stages  of  a  design,  it  is  usual  to  assume  that 
the  pole  shoes  are  so  shaped  as  to  give  a  sinusoidal  distribution 
of  flux  over  the  armature  surface.  The  calculation  of  a  correct- 
ing factor  for  distributed  windings  is  then  very  simple.  Thus, 
if  there  are  two  slots  per  pole  per  phase  in  a  three-phase  machine, 
there  will  be  six  slots  per  pole  pitch,  the  angular  distance  between 

180 
them  being  — ~-  =  30  electrical  degrees.     It  is  therefore  merely 

necessary  to  add  together,  vectorially,  two  quantities  having  a 
phase  displacement  of  30  degrees,  each  representing  the  e.m.f. 
developed  in  a  single  conductor.  The  result,  divided  by  2,  will 
be  the  average  voltage  per  conductor  of  the  distributed  winding. 
As  an  example,  with  three  slots  per  pole  per  phase,  the  graphic 
construction  would  be  as  indicated  in  Fig.  90  where  length 
AB  =  length  BC  =  length  CD,  and  what  [may  |be  called  the 

AD 

distribution  factor  is  k  =    040-     The  value   of  this   distribu- 


FIG.  90. — Vector  construction  to  determine  distribution  factor. 

tion  factor  is  therefore  always  either  equal  to,  or  less  than,  unity. 
If  Z  =  the  total  number  of  inductors  in  series  per  phase,  the 
final  formula  for  the  developed  voltage  is: 

E  (per  phase)  =  6Q  *^  Qg  X  form  factor  (93) 

On  the  sine  wave  assumption,  the  form  factor  is  1.11,  and  the 
formula  may,  if  preferred,  be  used  in  the  form 

E  (per  phase)  =  -     1fi8   '-  volts  (on  sine-wave  assumption)  (94) 

Values  of  k  can  easily  be  calculated  for  any  arrangement  of 
slots  and  windings.  With  a  full-pitch  three-phase  winding, 
the  distribution  factor,  k,  will  have  the  following  values: 

Number  of  slots  per  pole  Distribution  factor, 

per  phase  k 

1 1.0 

2 0.966 

3 0.960 

4 0.958 

Infinite. .  .0.955 


244 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


71.  Star  and  Mesh  Connections. — Consider  the  armature 
winding  of  an  ordinary  continuous-current  two-pole  dynamo. 
If  we  imagine  the  commutator  of  such  a  machine  to  be  entirely 
removed,  the  winding — whether  the  armature  be  of  the  drum 
or  ring  type — will  be  continuous,  and  closed  upon  itself.  If  the 
armature  be  revolved  between  the  poles  of  separately  excited 
field  magnets,  there  will  be  no  circulating  current  in  the  windings, 
because  the  magnetism  which  passes  out  of  the  armature  core 
induces  an  e.m.f.  in  the  conductors  exactly  opposite,  but  equal 

in  amount,  to  that  induced  by  the 
entering  magnetism. 

If  we  now  connect  two  points  of 
the  winding  from  the  opposite  ends 
of  a  diameter  to  a  pair  of  slip  rings, 
the  machine  will  be  capable  of  de- 
livering an  alternating  current.  If 
we  provide  three  slip  rings,  and 
connect  them  respectively  to  three 
points  on  the  armature  winding  dis- 
tant from  each  other  by  120  de- 
grees, the  machine  will  become  a 
three-phase  generator. 

In  this  manner  polyphase  cur- 
rents of  any  number  of  phases  can 
be  obtained,  and  if  the  windings 
and  field  poles  are  symmetrically 
arranged,  there  will  be  no  circu- 
lating current. 

This  method  of  connecting  up  the  various  armature  circuits 
of  a  polyphase  generator  is  known  as  the  mesh  connection.  In 
the  case  of  three-phase  currents  it  is  usually  referred  to  as  the 
delta  connection. 

The  diagram,  Fig.  91,  shows  the  three  equidistant  tappings 
from  armature  winding  to  slip  rings,  required  to  obtain  three- 
phase  currents.  It  is  evident  that  the  potential  difference  be- 
tween any  two  of  the  three  rings  will  be  the  same,  since  each 
section  of  the  winding  has  the  same  number  of  turns,  and  occupies 
the  same  amount  of  space  on  the  periphery  of  the  armature  core. 
Moreover,  the  variations  in  the  induced  e.m.f.  will  occur  suc- 
cessively in  the  three  sections  at  intervals  corresponding  to 
one-third  of  a  complete  period. 


FIG.  91. — Collection  of  three- 
phase  currents  from  bi-polar  ring 
armature. 


DESIGN  OF  ALTERNATORS 


245 


The  load  may  be  connected  across  one,  two,  or  three,  phases; 
but  in  practice,  especially  in  the  case  of  power  circuits,  the 
three-phase  load  is  usually  balanced',  i.e.,  each  phase  winding  of 
the  machine  provides  one-third  of  the  total  output. 


FIG.  92. — Diagram  of  connections  for  delta-connected  three-phase 
generator. 


Fig.  92  is  a  diagram  of  connections  referring  to  a  delta- 
connected  three-phase  generator,  and  Fig.  93  is  the  correspond- 
ing vector  diagram,  showing  how  the  current  in  the  external 
circuit  may  be  expressed  in  terms  of  the  armature  current.  The 
current  leaving  the  terminal 
A  (Fig.  92)  is  /i  -  /2,  and 
since  there  will  be  a  difference 
of  120  electrical  degrees  be- 
tween the  currents  I\  and  1 2, 
the  vector  construction  of  Fig. 
93  gives  01  as  the  line  cur- 
rent. Its  value  is  /  =  27a 
cos  30°,  or  x/3/a,  where  Ia  is 
the  current  in  the  armature 
conductors.  The  assumptions 
here  made  are  that  the  load 
is  balanced  and  that  the  cur- 
rent variations  follow  the  sim- 
ple harmonic  law.  It  is  well 


FIG.  93. — Vector  diagram  of  current 
relations  in  delta-connected  three-phase 
generator. 


to  bear  in  mind  that  vectors  and  vector  calculations  can  be  used 
only  when  the  variable  quantities  follow  the  sine  law;  when  used 
in  connection  with  irregular  wave  shapes,  they  must  be  supposed 
to  represent  the  " equivalent"  sine  function,  because  under  no 
other  condition  can  the  phase  angle  have  any  definite  meaning. 


246          PRINCIPLES  OF  ELECTRICAL  DESIGN 

Star  Connection  of  Three-phase  Armature  Windings. — If  the 
starting  ends  of  all  the  phase  windings  of  a  polyphase  generator 
are"  connected  to  a  common  junction,  or  neutral  point,  the  arma- 
ture windings  are  said  to  be  star-connected.  In  the  three- 
phase  machine  this  is  also  referred  to  as  the  Y  connection. 
The  outgoing  lines  being  merely  a  continuation  of  the  phase 
windings,  it  follows  that,  with  a  star-connected  machine,  the 
line  current  is  exactly  the  same  as  the  current  in  the  armature 
windings.  The  voltage  between  terminals  is,  however,  no  longer 
the  same  as  the  phase  voltage,  as  in  the  case  of  the  previously 
considered  mesh-connected  machine.  Referring  to  the  vector 
diagram  Fig.  94,  we  see  that  the  voltage  between  lines  1  and  2 
is  EI  —  E2,  which  leads  to  the  relation  E  =  \/3Ea  where  E  is 


FIG.  94. — Vector  diagram  showing  voltage  relations  in  Y-connected  three- 
phase  generator. 

the  terminal  voltage,  and  Ea  the  phase  voltage  as  measured 
between  any  one  terminal  and  the  neutral  point. 

There  is  little  to  be  said  in  regard  to  the  choice  of  armature 
connections  in  a  three-phase  generator,  except  that,  for  the 
higher  pressures,  the  Y  connection  has  the  advantage  of  a  lower 
voltage  per  phase  winding,  and,  for  heavy  current  outputs,  the 
A  connection  has  the  advantage  of  a  smaller  current  per  phase 
winding. 

Effect  of  Star  and  Delta  Connections  on  Third  Harmonic. — 
There  is  one  difference  resulting  from  the  method  of  connecting 
the  phase  windings  of  a  three-phase  generator  which  should  be 
mentioned.  This  has  reference  to  the  wave  shape  of  the  e.m.f. 
The  wave  shape  of  the  terminal  voltage  is  not  necessarily  the 


DESIGN  OF  ALTERNATORS  247 

same  as  that  of  the  e.m.f.  developed  in  the  armature  windings. 
Thus,  what  is  known  as  the  third  harmonic,  and  all  multiples 
of  the  third  harmonic,  are  absent  from  the  voltage  measured 
across  the  terminals  of  a  star-connected  three-phase  generator. 
By  the  third  harmonic  is  meant  a  sine  wave  of  three  times  the 
periodicity  of  the  fundamental  sine  wave,  which,  when  superim- 
posed on  this  fundamental  wave,  produces  distortion  of  the 
wave  shape. 

A  voltmeter  placed  across  the  terminals  of  a  star-connected 
generator  measures  the  sum  of  two  vector  quantities  with  a 
phase  difference  of  60  degrees  (see  Fig.  94).  Since  a  phase  dis- 
placement of  60  degrees  of  the  fundamental  wave  is  equivalent 
to  a  phase  displacement  of  60  X  3  =  180  degrees  of  the  third 
harmonic,  it  follows  that  the  third  harmonics  cancel  out  so  far 
as  their  effect  on  the  terminal  voltage  is  concerned.  The  general 
rule  is  that  the  nth  harmonic  and  its  multiples  cannot  appear 
in  the  terminal  voltage  of  a  star-connected  polyphase  generator 
of  n  phases.  The  same  arguments  apply  to  the  line  current  of  a 
raes/i-connected  polyphase  generator;  the  nth  harmonic  of  the 
current  wave  can  circulate  only  in  the  armature  windings;  it 
cannot  make  its  appearance  in  the  current  leaving  the  terminals 
of  the  machine. 

72.  Power  Output  of  Three-phase  Generator. — Let  E  and  / 
be  the  line  voltage  and  line  current,  and  let  Ea  and  Ia  stand  for 
the  phase  voltage  and  armature  current,  respectively;  then,  in 
the  A-connected  machine, 

E  =  Ea 
and 

/    =    \/3/« 

while,  in  the  Y-connected  machine, 

E  =  V3Ea 
and 

/    =    /a 

Assuming  unity  power  factor,  we  may  write: 
Output  of  A-connected  machine  =  3(EaIa) 

=  3EX 


V3 

=  VZEI 

and,  similarly, 


248          PRINCIPLES  OF  ELECTRICAL  DESIGN 

Output  of  Y-connected  machine  =  3(EaIa) 

=  3-^X7 

V3 


The  total  output  is,  of  course,  the  same  in  both  cases.  If  the 
power  factor  is  not  unity,  the  output,  in  watts,  of  the  three- 
phase  generator  on  a  balanced  load  is: 

W  =  \/3EI  cos  6. 

where  8  is  the  angle  of  lag  between  terminal  e.m.f  .  of  a  phase 
winding  and  current  in  the  winding.  The  quantity  cos  6  is  the 
power  factor  when  both  current  and  e.m.f.  waves  are  sinusoidal. 

Since  the  magnetic  circuit  of  an  alternating-current  generator 
has  to  be  designed  for  a  certain  flux  to  develop  a  given  voltage, 
while  the  copper  windings  must  be  of  sufficient  cross-section  to 
carry  a  given  current,  the  size  of  the  machine  will  depend  upon 
the  product  of  volts  and  amperes,  and  not  upon  the  actual 
power  output.  Alternating-current  generators  are  therefore 
rated  in  kilovolt-amperes  (k.v.a.),  the  actual  output,  in  kilowatts, 
being  dependent  upon  the  power  factor  of  the  external  circuit. 

73.  Usual  Voltages.  —  Owing  to  the  absence  of  the  commu- 
tator, A.C.  machines  can  be  wound  for  higher  voltages  than 
D.C.  machines.     Large  A.C.  generators  may  be  wound  to  give 
as  high  a  pressure  as  16,000  volts  at  the  terminals,  but  it  is 
rarely  economical  to  develop  much  above  13,000  volts  in  the 
generator;  when  higher  pressures  are  required,  as  for  long  dis- 
tance power  transmission,  step-up  transformers  are  used.     In 
this  country,  a  very  common  terminal  voltage  for  three-phase 
generators  to  be  used  in  connection  with  step-up  transformers, 
is  either  2,200  or  6,600  volts,  the  higher  voltage  being  adopted 
for  the  larger  outputs,  in  order  to  avoid  heavy  currents  in  the 
machine  and  between  the  machine  and  the  primary  terminals 
of  the  transformer. 

74.  Pole  Pitch  and  Pole  Arc.  —  Although  there  can  be  no 
sparking  at  the  sliding  contacts,  as  in  D.C.  designs,  with  their 
commutation  difficulties,  the  effects  of  field  distortion  and  demag- 
netization are  apparent  in  the  voltage  regulation  of  alternating- 
current  generators.     A  very  large  pole  pitch,  involving  as  it 
does  a  large  number  of  ampere-conductors  per  pole,  is  objec- 
tionable, and  should  be  avoided  if  possible.     Where  it  is  un- 
avoidable, a  large  air  gap  must  be  provided  in  order  to  prevent 


DESIGN  OF  ALTERNATORS  249 

the  armature  m.m.f.  overpowering  the  field  excitation;  but  this 
leads  to  increased  cost. 

The  pole  pitch,  r,  is  a  function  of  the  peripheral  speed  and 

the  frequency,  thus: 

_7rZ) 

P 

where  p  =  number  of  poles,   and  D  =  diameter  of  armature 
(inches). 
But 


2 

and 

pN 


f  = 

ipheral  veloc 
It  follows  that 


120 
where  v  =  peripheral  velocity  of  armature  in  feet  per  minute. 


(95) 


which  explains  why  the  pole  pitch  is  always  large  in  steam- 
turbine-driven  alternators. 

In  60-cycle  machines  running  at  moderate  speeds,  the  pole 
pitch  usually  lies  between  6  and  12  in.,  a  pitch  of  8  to  10  in. 
being  very  common.  The  speed  of  the  machine  —  which  is 
usually  a  factor  in  determining  the  peripheral  velocity  —  has 
an  appreciable  influence  upon  the  choice  of  pole  pitch;  pole  cores 
of  approximately  circular  or  square  section  are  not  always  feasible 
or  economical,  and  the  designer  must  make  some  sort  of  a  com- 
promise to  get  the  best  proportions.  The  output  formula,  as 
used  for  determining  the  proportions  of  dynamos,  is  not  so 
readily  applicable  to  the  design  of  alternators,  because  the  arma- 
ture diameter  in  A.C.  machines  will  be  settled  largely  by  con- 
siderations of  peripheral  velocity  and  pole  pitch,  the  propor- 
tions of  the  pole  face  being  a  secondary  matter.  The  value  of 

k.  or  ratio  -      P,   6    5  —  rr,  is  therefore  determined  largely  by 
armature  length' 

the  limits  of  peripheral  velocity,  which  may  lead  to  a  smaller 
diameter  and  greater  axial  length  than  would  be  strictly  eco- 
nomical if  the  weight  of  copper  in  the  field  coils  were  the  only 
consideration. 

It  is  usual,  when  possible,  to  limit  the  armature  ampere-turns 
per  pole  to  10,000,  which  will  determine  the  maximum  permis- 


250          PRINCIPLES  OF  ELECTRICAL  DESIGN 

sible  value  of  the  pole  pitch;  but  this  limit  must  sometimes 
be  exceeded,  as  in  the  case  of  steam-turbine-driven  machines, 
in  which  a  pole  pitch  of  3  to  4  ft.  is  by  no  means  uncommon. 

The  pole  arc  is  a  smaller  portion  of  the  total  pitch  than  in 
continuous-current   machines;   the   value    of   r    (i.e.,  'the   ratio 

~  rarely  exceeds  0.65.     A  very  common  value  is  0.6, 


while  it  may  frequently  be  as  low  as  0.55.  The  reason  for 
the  smaller  circumferential  space  occupied  by  the  pole  face  is 
partly  to  avoid  excessive  magnetic  leakage,  but  mainly  to  pro- 
vide a  proper  distribution  of  flux  over  the  pole  pitch.  An  at- 
tempt is  usually  made  to  obtain  sinusoidal  distribution;  but  the 
means  of  obtaining  this  will  be  explained  later. 

75.  Specific  Loading.  —  As  in  the  case  of  D.C.  machines,  the 
specific  loading,  q,  is  defined  as  the  number  of  ampere-conductors 
per  inch  of  armature  periphery.  The  conductors  of  all  phases 
are  counted,  and  the  current  considered  is  the  virtual,  or  r.m.s., 
value  of  the  armature  current.  The  magnetizing  effect  of  the 
armature  as  a  whole  will,  at  any  moment,  depend  upon  the 
instantaneous  value  of  the  currents  in  the  individual  conductors, 
but  this  matter  will  be  taken  up  later. 

The  following  are  average  values  of  q,  as  found  in  commercial 
machines  : 

Outpjit  of  A.C.  gener-  Average  value 

ator  (k.v.a.)  of  q 

50  400 

100  430 

200  470 

500  520 

1,000  570 

5,000  .        625 

10,000  670 

The  proper  value  of  q  to  be  used  in  a  given  design  will  depend 
on  several  factors.  Apart  from  the  fact  that  its  value  increases 
with  the  size  of  the  machine,  it  will  depend  somewhat  upon  the 
following  factors: 

(a)  Number  of  poles. 

(6)  Frequency. 

(c)  Voltage. 

(a)  Machines  with  a  small  number  of  poles  usually  have  a 
small  armature  diameter  and  a  large  pole  pitch;  calling  for  a 
small  value  of  q.  In  modern  steam-turbine-driven  generators 


DESIGN  OF  ALTERNATORS  251 

there  is,  however,  a  tendency  to  use  high  values  of  q  in  order  to 
limit  the  length  of  the  armature  (and  increase  the  critical  speed) 
and  also  to  increase  the  armature  m.m.f.  with  a  view  to  lowering 
the  short-circuit  current. 

(6)  With  low  frequency  it  is  easy  to  keep  the  iron  loss  small, 
and  more  copper,  or  a  greater  current  density  in  the  conductors, 
is  permissible. 

(c)  If  the  e.m.f.  is  low,  the  insulation  occupies  less  space, 
and  there  is  more  room  for  copper  without  unduly  reducing  the 
cross-section  of  the  armature  teeth. 

The  approximate  figures  given  in  the  above  table  may  be 
increased  or  reduced  about  20  per  cent.,  the  highest  values  being 
used  only  when  there  is  a  combination  of  low  voltage,  low  fre- 
quency, and  large  number  of  poles. 

76.  Flux  Density  in  Air  Gap. — Since  the  pole  shoe  is  shaped 
to  give  as  nearly  as  possible  a  sinusoidal  distribution  of  flux 
density  over  the  pole  pitch,  it  is  convenient  to  think  of  the 
maximum  value  of  the  air-gap  density,  because  this  will  determine 
the  maximum  density  in  the  iron  of  the  teeth.  The  frequency 
being  usually  higher  than  in  B.C.  machines,  lower  tooth  densities 
must  be  used  in  order  to  avoid  excessive  loss.  The  allowable 
flux  density  in  the  air  gap  will  depend  upon  the  proportions  of 
tooth  and  slot;  but  the  following  values  may  be  used  for  pre- 
liminary calculations. 

For  a  frequency  of  25,  Ba  =  4,000  to  5,800  gausses. 
For  a  frequency  of  60,  B0  =  3,500  to  5,000  gausses. 

These  values  of  Bg  stand  for  the  average  density  over  ihe  pole 
pitch.  If  $  =  the  total  number  of  maxwells  per  pole;  and  the 
shape  of  the  flux  distribution  over  the  pole  pitch  is  assumed  to 
be  a  sine  wave,  we  have: 

$ 
Area  of  pole  pitch  =  TT 

£>a 

which  determines  the  axial  length  of  the  armature  core.  The 
maximum  air-gap  density,  on  the  above  assumption,  is  ^  Bg,  and 

after  deciding  upon  the  tooth  and  slot  proportions,  it  is  advisable 
to  see  that  this  density  will  not  lead  to  an  unreasonable  value 
for  the  apparent  tooth  density.  As  a  check,  it  may  be  stated 
that  the  tooth  density  in  alternators  is  rarely  higher  than  18,000 
gausses  in  25-cycle  machines,  and  16,000  gausses  in  60-cycle 


252          PRINCIPLES  OF  ELECTRICAL  DESIGN 

machines.  Higher  densities  are  used  in  some  steam-turbine- 
driven  machines,  with  a  view  to  reducing  the  size  of  the  rotor. 
A  good  system  of  forced  ventilation  is  then  imperative. 

77.  Length  of  Air  Gap. — Inherent  Regulation. — In  A.C.  ma- 
chines, just  as  in  D.C.  machines,  the  length  of  air  gap  should 
depend  upon  the  armature  m.m.f  and  therefore  on  the  pole 
pitch,  T,  and  the  specific  loading,  q.  In  salient-pole  machines, 
the  air  gap  will  not  be  of  constant  length,  but  will  increase  from 
the  center  outward,  in  order  to  produce  the  required  distribution 
of  flux.  A  practical  method  of  shaping  the  pole  face  will  be 
explained  later.  The  clearance  to  be  allowed  between  pole 
face  and  armature  surface  at  the  center  of  the  pole  may  be  de- 
termined approximately  by  making  it  of  such  a  length  that  the 
open-circuit  field  ampere-turns  shall  be  not  less  than  1.75  times 
to  twice  the  full-load  armature  ampere-turns.  In  large  turbo- 
alternators  this  ratio  may  be  as  low  as  1  to  1.5,  in  order  to  reduce 
the  weight  of  copper  on  the  rotor,  and  keep  the  short-circuit 
current  within  reasonable  limits.  The  distribution  of  armature 
m.m.f.  will  be  discussed  later;  but,  for  the  purpose  of  estimating 

the  air  gap,  the  ampere-turns   per   pole  may  be  taken  as  -*- 

In  no  case  should  the  air  gap  be  less  than  one-third  to  one-half 
the  slot  opening. 

A  large  air  gap  has  the  effect  of  improving  the  regulation  of 
the  machine;  but  otherwise  it  is  objectionable,  seeing  that  it 
leads  to  increased  magnetic  leakage  and  higher  cost,  due  mainly 
to  the  greater  weight  of  copper  in  the  field  coils. 

The  inherent  regulation  of  a  generator,  at  any  given  load,  may 
be  defined  as  the  percentage  increase  in  terminal  voltage  when 
the  load  is  thrown  off;  the  speed  and  field  excitation  remaining 
constant.  Owing  to  the  low  power  factors  resulting  from  the 
connection  of  induction  motors  on  alternating-current  circuits, 
it  is  practically  impossible  to  design  a  generator  of  which  the 
inherent  regulation  is  so  good  that  auxiliary  regulating  devices 
are  unnecessary.  It  is,  therefore,  uneconomical  to  aim  at  very 
good  inherent  regulation,  especially  as  efficient  automatic  field 
regulators  are  now  available.  The  inherent  regulation  of  com- 
mercial machines  usually  lies  between  5  and  9  per  cent,  at  full 
load  on  unity  power  factor,  while  it  may  easily  be  20  per  cent., 
or  higher,  on  85  per  cent,  power  factor,  with  normal  full-load 
current  taken  from  the  machine.  This  very  marked  effect  of 


DESIGN  OF  ALTERNATORS  253 

a  low  power  factor  will  be  explained  later  in  detail;  but  it  may 
be  stated  here  that  the  effect  of  a  lagging  armature  current 
is  very  similar  to  that  of  a  change  of  brush  position  in  a  con- 
tinuous-current dynamo,  causing  the  armature  ampere-turns — 
which  on  unity  power  factor  have  merely  a  distorting  effect — 
to  become  partly  demagnetizing. 

Not  only  must  the  armature  be  weak  relatively  to  the  field, 
but  the  inductance  of  the  armature  windings  should  be  small 
if  the  inherent  regulation  is  to  be  good.  Thus  the  regulating 
qualities  of  an  alternating-current  generator  depend  on  both 
armature  reaction  and  armature  reactance;  but  since  these  cannot 
be  made  so  small  as  to  dispense  entirely  with  external  regulating 
devices,  the  designer  rarely  aims  at  getting  very  good  inherent 
regulation. 

With  steam-turbine-driven  machines,  in  which  the  pole  pitch 
is  always  very  large,  the  air  gap  frequently  exceeds  1  in.  in 
length.  The  writer  knows  of  a  machine,  designed  for  an  output 
of  5,500  k.v.a.  at  a  speed  of  1,000  revolutions  per  minute,  and 
a  frequency  of  33J£,  with  the  single  air  gap  3^  in-  long.  Whether 
or  not  the  designer  was  justified  in  trying  to  obtain  satisfactory 
regulation  by  this  costly  and  somewhat  crude  expedient  is  at 
least  questionable.  Machines  of  three  times  this  output  arc 
now  built  with  air  gaps  from  1  in.  to  lj^  in.  long. 

Good  inherent  regulation  means  that  the  current  on  short- 
circuit  may  be  very  large,  and  this  is  sometimes  objectionable. 
With  the  exception  of  high-speed,  steam-turbine-driven  units, 
the  short-circuit  current  in  modern  A.C.  generator  (with  full- 
field  excitation)  is  about  three  to  five  times  the  normal  full-load 
current;  but  in  connection  with  the  larger  units,  and  on  systems 
dealing  with  large  amounts  of  energy,  power-limiting  reactances, 
external  to  the  generator,  are  usually  installed  to  prevent  the 
current  attaining  a  dangerous  value  before  the  automatic  circuit- 
breakers  have  had  time  to  operate.  Many  of  the  largest  units, 
driven  at  very  high  speeds  by  steam  turbines,  are  now  purposely 
designed  with  large  armature  reaction  and  highly  inductive 
windings,  in  order  that  they  may  be  able  to  withstand  momen- 
tary short-circuits  without  mechanical  injury;  but  notwith- 
standing these  features  of  recent  introduction,  the  momentary 
short-circuit  current  in  some  of  the  20,000  to  30,000  k.v.a.  units, 
may  be  of  the  order  of  15  to  20  times  the  normal  full-load  current. 

For  certain  electro-metallurgical  work,  or  electric  smelting, 


254          PRINCIPLES  OF  ELECTRICAL  DESIGN 

as,  for  instance,  the  electric  production  of  calcium  carbide, 
an  alternator  with  poor  regulation  is  desirable.  In  other  words, 
where  a  constant-power  machine  is  needed,  a  powerful  armature 
reaction  and  magnetic  leakage  are  useful;  with  a  decrease  in  the 
resistance  of  an  electric  furnace,  the  current  will  rise,  but  if  this 
increase  of  current  causes  a  falling  off  in  the  pressure  at  the 
generator  terminals,  the  power  consumed  will  not  increase  to  any 
appreciable  extent. 


CHAPTER  XII 

ARMATURE  WINDINGS— LOSSES,  AND  TEMPERATURE 

RISE 

78.  Types  of  Windings. — Fundamental  winding  diagrams  for 
single-,  two-,  and  three-phase,  machines  were  illustrated  and  ex- 
plained in  the  preceding  chapter  (Art.  67).  Beyond  this  it  is 
not  proposed  to  say  much  regarding  the  actual  arrangement  of 
armature  windings  in  alternating-current  generators.  Much 
excellent  matter  has  been  published  on  the  practice  of  armature 
winding;1  but  it  has  little  to  do  with  the  principles  of  electric 
design,  and,  in  the  end,  is  really  a  study  of  the  most  convenient 
and  economical  way  of  connecting  together  the  active  conductors 
in  the  slots.  There  is  almost  no  limit  to  the  number  of  styles 
of  winding  that  can  be  used  on  alternators,  or  to  the  names  that 
may  be,  and  are,  given  to  these  different  windings;  but  the  funda- 
mental principles  underlying  the  generation  of  an  alternating 
e.m.f.  can  be  studied  without  a  detailed  knowledge  of  the  many 
practical  types  of  armature  windings. 

There  is  one  broad  distinction  that  can  be  made,  and  alternator 
windings  may  be  divided  into : 

(a)  Double-layer  windings. 

(6)  Single-layer  windings. 

(a)  Double-layer  Winding. — This  is  practically  identical  with 
the  usual  D.C.  winding,  the  coils  being  generally  of  the  same 
shape;  but  instead  of  tappings  being  taken  to  a  commutator, 
the  coils  are  connected  together  in  the  proper  order,  the  phase 
windings  being  kept  separate  until  finally  connected  star  or  mesh 
as  may  be  decided.  With  this  style  of  winding,  the  number  of 
conductors  per  slot  must  be  a  multiple  of  two.  All  coils  are  of 
the  same  size  and  shape,  which  is  an  advantage;  but  on  the  other 
hand,  the  end  connections  are  rather  close  together,  and  there 
must  also  be  substantial  insulation  between  the  two  coil-sides 
in  each  slot.  This  type  of  winding  is,  therefore,  not  very  suitable 
for  high  pressures.  One  great  advantage  of  the  double-layer 

1  MILES  WALKER:  "Specification  and  Design  of  Dynamo-electric  Ma- 
chines," LONGMANS  &  Co. 

255 


256 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


winding  is  that  it  lends  itself  readily  to  fractional  pitch  lap 
windings,  in  which  the  two  sides  of  a  coil  are  not  similarly  placed 
relatively  to  the  center  lines  of  the  poles,  with  the  result  that  tooth 
harmonics  in  the  e.m.f.  wave  may  be  almost  eliminated. 

(6)  Single-layer  Winding. — With  this  winding  there  is  only 
one  coil-side  in  a  slot,  and  the  number  of  conductors  per  slot 
may,  therefore,  be  either  odd  or  even.  Several  shapes  of  coil  are 


FIG.  95. — Three-phase,  single-layer  winding;  three  slots  per  pole  per  phase. 

necessary  in  order  that  the  end  connections  may  clear  each  other, 
and  this  involves  a  larger  number  of  special  tools  or  formers  and 
a  larger  number  of  spare  coils  than  for  a  double-layer  winding. 
These  disadvantages  are,  however,  sometimes  outweighed  by  the 
fact  that  the  total  number  of  coils  in  the  machine  is  smaller. 
Good  insulation  is  easily  obtained  because  the  end  connections 
may  be  separated  by  large  air  spaces,  and  the  single-layer  winding 
is,  therefore,  suitable  for  high  voltages. 


FIG.  96. — Three-phase,  single  layer  winding;  four  slots  per  pole  per  phase 

Considering  each  phase  winding  separately,  the  full  number  of 
turns  per  pole  may  encircle  one  pole  only  as  shown  in  Fig.  95,  or 
they  may  be  divided  between  a  pair  of  poles,  in  two  equal  parts, 
as  shown  in  Fig.  96.  Both  diagrams  show  one  phase  only  of  a 
three-phase  generator.  In  Fig.  95  there  are  three  slots,  while  in 
Fig.  96  there  are  four  slots,  per  pole  per  phase.  The  coils  of  the 
other  phase  windings  would  be  similarly  arranged  in  the  re- 
maining slots,  the  ends  projecting  beyond  the  slots  being  shaped 


ARMATURE  WINDINGS 


257 


or  bent  so  as  to  clear  the  other  coils,  generally  as  shown  in, 
Fig.  97. 

All  the  conductors  of  one  phase  are  usually  connected  in 
series,  but  sometimes  parallel  circuits  are  used.  It  then  becomes 
a  matter  of  importance  to  see  that  there  is  no  phase  difference 
between  the  e.m.fs.  generated  in  the  conductors  of  parallel 
circuits.  In  other  words,  the  conductors  of  parallel  circuits 
should  be  so  disposed  in  the  available  slots  that  they  cut  the 
same  amount  of  flux  at  the  same  instant  of  time.  Having 
mentioned  this  point,  it  does  not  appear  necessary  to  enlarge 
upon  it. 


FIG.  97. — End  connections  of  single-layer  armature  winding. 


79.  Spread  of  Windings. — In  two-phase  and  three-phase 
machines,  all  the  slots  on  the  armature  are  utilized.  With 
full-pitch  windings,1  the  number  of  slots  per  pole  is  divisible  by 
2  for  a  two-phase  generator,  and  by  3  for  a  three-phase  generator. 
Thus,  with  distributed  windings  (more  than  one  slot  per  pole 
per  phase),  the  "spread,"  or  space  occupied  by  each  phase  wind- 
ISO 
ing,  is  -g-  =  90  electrical  degrees  for  a  two-phase  machine,  and 

180 

-Q-  =  60  degrees  for  a  three-phase  machine. 

In  single-phase  machines,  nothing  is  gained  by  winding  all 
the  slots  on  the  armature  surface;  after  a  certain  width  of  wind- 
ing has  been  reached,  the  filling  of  additional  slots  merely  in- 
creases the  resistance  and  inductance  of  the  winding,  without  any 
appreciable  gain  in  the  matter  of  developed  voltage.  This  is 

1  Short-pitch  windings  are  very  common  in  two-pole  machines,  as  they 
tend  to  simplify  the  end  connections.  In  this  case  the  double-layer  winding, 
as  in  D.C.  machines,  would  be  used. 

17 


258  PRINCIPLES  OF  ELECTRICAL  DESIGN 

made  clear  in  the  vector  diagram,  Fig.  98.  The  winding  is  sup- 
posed to  be  distributed  in  a  very  large  number  of  slots,  and  the 
diameter  of  the  semicircle  represents  the  resultant  generated 
e.m.f.  if  all  the  slots  are  filled  with  conductors  (connected  in 
series).  If,  as  is  usual  in  practice,  only  75  per  cent,  of  the  slots 
are  utilized,  the  spread  of  the  single-phase  winding  will  be  about 
135  electrical  degrees;  the  resultant  e.m.f.  will  be  AB,  which  is 
not  much  shorter  than  AC',  but  the  length  and  weight  of  copper 

arc  AB 

in  the  two  cases  are  in  the  proportion  -       A  Kr<  -     The  fact  that, 

arc 


in  polyphase  machines,  the  whole  of  the  armature  surface  is 
available  for  the  windings,  while  only  a  portion  of  this  surface 
is  utilized  in  the  single-phase  alternator,  accounts  for  the  fact 
that  the  output  of  the  latter  is  less  than  that  of  the  polyphase 
machine  for  the  same  size  of  frame.  Given  a  three-phase  machine, 
it  is  merely  necessary  to  omit  one  of  the  phase  windings  entirely 


FIG.  98. — Vector  diagram  illustrating  "spread"  of  armature  winding  in 
single-phase  alternator. 

'and  connect  the  two  remaining  phases  in  series,  to  obtain  a 
single-phase  generator.  The  modified  machine  will  be  capable 
of  giving  something  more  than  two-thirds  of  the  output  of  the 
polyphase  generator,  the  limit  being  reached  when  the  copper 
losses  become  excessive. 

80.  Insulation  of  Armature  Windings. — With  very  high 
voltages,  such  as  are  used  on  many  power  transmission  schemes, 
a  special  study  has  to  be  made  of  the  problems  of  insulation. 
These  problems  then  become  of  extreme  importance,  and  many 
difficulties  have  to  be  overcome  that  do  not  trouble  the  designer 
who  is  dealing  with  pressures  of  the  order  of  5,000  to  10,000 
volts. 

The  reader  is  referred  to  Art.  28  of  Chap.  V,  where  slot  insula- 
tion was  discussed,  and  since  the  same  insulating  materials 
are  used  in  alternators  as  in  dynamos,  there  is  little  to  be  added 
here.  It  is  the  practice  of  some  manufacturers  to  have  the 


ARMATURE  WINDINGS  259 

question  of  insulation  studied  by  experts  who  decide  upon  the 
most  suitable  materials  to  withstand  the  particular  conditions 
under  which  the  machine  will  have  to  operate,  and  then  advise 
the  designer  of  the  machine  regarding  the  space  to  be  allowed 
to  accommodate  this  insulation.  If  high-class  insulating  materials 
are  used,  the  slot  lining,  i.e.,  the  total  thickness  of  insulation 
between  the  conductors  and  the  side  or  bottom  of  the  slot, 
should  have  the  following  values: 


Terminal  voltage 

500  0.045  in. 

1,000  0.060  in. 

2,000  0.080  in. 

4,000  0.12    in. 

8,000  0.19    in. 

12,000  0.27    in. 

81.  Current  Density  in  Armature  Conductors.  —  Although  the 
armature   may  be  stationary,  the   permissible  current   density 
in  the  conductors  will  depend  to  some  extent  upon  the  peripheral 
speed  of  the  rotating  field  magnets,  because  the  fanning  effect 
will  be  greater  at  the  higher  velocities.     The  cooling  effect  of 
the  air  thrown  against  the  conductors  by  the  rotation  of  the 
field  magnets  is  not  so  great  as  when  the  armature  rotates,  and 
moreover,  the  air  is  warmed  to  some  extent  in  passing  over  the 
heated  surface  of  the  field  coils.     The  current  density  in  alter- 
nator armatures  usually  lies  between  1,500  and  3,000  amp.  per 
square  inch  of  cross-section.     The  formula  previously  used  in 
the  design  of  dynamo  armatures  requires  some  modification, 
and  the  writer  proposes  the  following  empirical  formula  for 
current  density  in  the  armature  windings  of  alternators  with 
rotating  field  system,  up  to  a  peripheral  speed  of  8,000  ft.  per 
minute  : 

600,000       v 
A  =      —  •  +  g  (96) 

The  symbols  have  the  same  meaning  as  in  formula  (51)  on  page 
97;  the  peripheral  velocity,  v,  being  calculated  by  assuming 
that  the  armature  is  rotating  instead  of  the  field. 

82.  Tooth    and    Slot    Proportions.  —  In    deciding    upon    the 
number   of   teeth   on   the   armature,    a   compromise  must  be 
made  between  a  very  small  number  of  teeth  —  which  involves 
the   bunching   of   conductors,    with   consequent   high   internal 


260          PRINCIPLES  OF  ELECTRICAL  DESIGN 

temperatures  and  high  inductance — and  a  large  number  of  teeth, 
involving  more  space  taken  up  by  insulation,  and  a  higher  cost 
generally.  Although  larger  slots  are  permissible  in  A.C.  than 
in  D.C.  machines,  a  slot  pitch  (X)  greater  than  2.5  in.  is  not 
recommended.  The  upper  limit  might  be  2.75  in.  if  the  air 
gap  is  large,  while  the  lower  limit  is  determined  by  considera- 
tions of  space  available  for  conductors  and  insulation,  bearing  in 
mind  the  higher  cost  of  a  large  number  of  coils.  In  large  turbo- 
alternators,  the  slot  pitch  may  be  as  large  as  3  in.  or  even 
3^,  in.  but  in  such  cases  a  slot  wedge  built  up  of  laminated 
iron  plates  is  generally  used,  thus  virtually  reducing  the  slot 
opening  and  equalizing  the  flux  distribution  over  the  slot  pitch. 
In  a  three-phase  machine,  the  number  of  slots  per  pole  per  phase 
is  usually  from  1  to  4;  but  in  turbo-alternators,  with  large  pole 
pitch,  the  number  of  slots  may  greatly  exceed  these  figures. 

The  conductors  must  be  so  arranged  that  the  width  of  slot 
is  not  such  as  to  reduce  the  tooth  section  beyond  the  limit 
corresponding  to  a  reasonable  flux  density  in  the  iron  of  the 
tooth  (see  Art.  76  of  the  preceding  chapter);  but,  on  the  other 
hand,  a  deep  slot  is  sometimes  objectionable  because  it  leads 
to  a  high  value  of  slot  leakage  flux.  The  depth  of  the  slot 
should  preferably  not  exceed  three  times  the  width,  although 
deeper  slots  can  be  used,  and  may,  indeed,  be  desirable  in  cases 
where  poor  inherent  regulation  is  deliberately  sought. 

83.  Length  and  Resistance  of  Armature  Winding. — Apart 
from  the  pitch,  T,  and  the  gross  length  of  armature  core,  la,  the 
length  per  turn  of  the  winding  will  depend  upon  the  voltage  and 
also  upon  the  slot  dimensions.  The  voltage  will  determine  the 
amount  by  which  the  slot  insulation  should  project  beyond  the 
end  of  the  armature  core,  and  the  cross-section  of  the  coil  will 
be  a  factor  in  determining  the  length  taken  up  in  bends  at  the 
corners  of  the  coil.  A  rough  sketch  of  the  coil  should  be  made, 
and  the  length  of  a  mean  turn  estimated  as  closely  as  possible 
for  the  purpose  of  calculating  the  resistance  and  weight.  With 
the  high  pressures  generated  in  some  machines,  it  is  necessary 
to  carry  the  slot  insulation  a  considerable  distance  beyond  the 
end  of  the  slot  in  order  to  guard  against  surface  leakage,  and 
although  no  definite  rules  can  be  laid  down  to  cover  all  styles 
of  winding,  the  straight  projection  of  the  coil-side  (and  insula- 
tion) outside  the  slot  would  be  at  least  %  (k.v.  +  1)  in.;  where 
k.v.  stands  for  the  pressure  between  terminals  in  kilovolts.  On 


ARMATURE  WINDINGS  261 

the  basis  of  an  average  size  of  slot,  the  actual  overhang  beyond 
end  of  core  would  have  a  mean  value  of  about  J^  (k.v.  +  3  +  j), 

where  r  is  the  pole  pitch  in  inches.     On  this  basis,  and  as  a  very 
rough  estimate,  the  mean  length  per  turn  in  inches,  would  be 

2/a  +  2.5r  +  2  k.v.  +  6  (97) 

The  cross-section  having  been  previously  decided  upon,  the 
resistance  per  phase  of  the  armature  winding  can  readily  be 
calculated. 

84.  Ventilation. — The  gross  length  of  the  armature  core  (la 
in  the  last  formula)  will  depend  upon  the  space  taken  up  by  the 
radial  vent  ducts  and  insulation  between  stampings.     If  radial 
ducts  are  used,  they  are  from  %  to  %  in.  wide,  spaced  2  to  4 
in.  apart,  the  closer  spacing  being  used  when  the  axial  length 
of  the  core  is  great  and  the  peripheral  velocity  low. 

In  turbo-alternators,  axial  vent  ducts  are  being  used  in  place 
of  radial  ducts.  If  there  are  no  radial  openings  between  the 
armature  plates,  the  length  of  the  core  can  be  reduced,  and  this 
is  always  desirable  in  high-speed  machines.  The  relation  be- 
tween net  and  gross  lengths  of  armature  core  will  then  be 
approximately  ln  =  0.92/a.  Even  when  axial  ducts  are  used, 
one  or  more  radial  openings  at  the  center  of  the  core  are  some- 
times provided  so  that  the  cool  air  may  be  drawn  in  at  both  ends 
of  the  armature  and  discharged  at  the  center.  The  fan  for  forced 
ventilation  may  be  inside  or  outside  the  generator.  In  large 
units  the  external  fan  is  generally  to  be  preferred.  The  reader 
is  referred  to  Art.  33,  Chap.  VI,  where  the  ventilation  of  dynamos 
was  discussed. 

85.  Full-load  Developed  Voltage. — The  losses  in  the  armature 
core  at  full  load  will  depend  upon  the  developed  e.m.f.,  which 
is  not  quite  so  easily  calculated  as  in  the  case  of  a  D.C.  dynamo. 
The  pressure  that  has  to  be  generated  in  the  armature  windings 
of  an  alternator  for  a  given  terminal  voltage,  will  depend  not 
only  upon  the  IR  pressure  drop,  but  also  on  the  IX  drop.     In 
other  words,  the  inductance  of  the  armature  windings,  and  the 
power  factor  of  the  load,  must  be  taken  into  account  when 
calculating  the  developed  voltage. 

The  vector  diagram,  Fig.  99,  refers  to  a  machine  working  on  a 
load  of  unity  power  factor.  The  current  is  in  phase  with  the 
terminal  voltage  OEt;  but  the  developed  volts  are  OEg  and  not 


262 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


OP  as  would  be  the  case  if  the  IR  drop  only  had  to  be  considered. 
The  vector  PE0  represents  the  e.m.f.  component  necessary  to 
counteract  the  reactance  drop  in  the  armature  windings.  Al- 
though the  external  power-factor  angle  is  zero,  there  is  an  angle 
\l/  between  the  current  vector  and  the  vector  of  the  developed 
e.m.f.,  which  may  be  termed  the  internal  power-factor  angle. 

Ea 


(IX  Drop) 


""-Internal  Power  Factor 

Angle 

FIGL  99. — Vector  diagram  for  calculating  developed  e.m.f. — non-inductive 

load. 

In  Fig.  100,  the  external  power-factor  angle  is  6  (power  factor 
of  load  =  cos  6).  The  construction  shows  how  the  reactance 
voltage  (IX)  becomes  a  factor  of  greater  importance  on  the  lower 
power  factors.  The  e.m.f.  that  must  be  developed  to  obtain 
a  constant  terminal  pressure  must,  therefore,  be  greater  on  low 
power  factor.  This,  however,  is  not  the  chief  cause  of  poor 
regulation  on  low  power  factors;  it  is  the  demagnetizing  effect  of 


(IX) 


FIG.  100. — Vector  diagram  for  calculating  developed    e.m.f. — load   partly 

inductive. 

the  armature  ampere-turns  which  is  chiefly  accountable  for 
poor  regulation  on  all  but  unity  power  factor.  From  an  in- 
spection of  Fig.  100  it  will  be  seen  that  the  greatest  difference 
between  developed  and  terminal  voltage  occurs  when  the  ex- 
ternal power-factor  angle  is  the  same  as  the  angle  EgEtP,  be- 
cause the  developed  voltage  is  then  simply  the  arithmetical  sum 
of  the  terminal  voltage  OEt  and  the  impedance  drop  E0Et. 


ARMATURE  WINDINGS  263 

In  connection  with  the  predetermination  of  temperature  rise, 
the  losses  in  the  armature  core  may  well  be  calculated  on  the 
assumption  that  this  condition  is  fulfilled. 

The  vector  diagrams  should  always  be  drawn  to  show  the 
relation  of  the  variable  quantities  in  one  phase  of  the  winding;  a 
balanced  load  being  assumed.  It  does  not  then  matter  whether 
the  phases  are  star-  or  delta-connected,  except  that,  in  the  case 
of  a  star-connected  generator,  the  vector  OEt  would  stand  for 
the  voltage  between  one  terminal  and  the  neutral  point,  and  its 

numerical    value    would    therefore   be    ~77*   times    the    voltage 

between  terminals. 

The  length  of  the  vector  EtP  in  Figs.  99  and  100  is  easily 
calculated;  but  the  numerical  value  of  IX  (the  vector  PE0) 
is  not  so  easily  estimated.  Consider  first  what  is  to  be  under- 
stood by  the  term  armature  reactance. 

86.  Inductance  of  A.C.  Armature  Windings. — It  is  not  always 
easy  to  separate  armature  reactance  (X)  from  armature  reac- 
tion (the  demagnetizing  effect  of  the  armature  ampere-turns). 
Both  cause  a  drop  of  pressure  at  the  terminals  under  load,  espe- 
cially on  low  power  factors.  By  departing  from  the  conventional 
methods  of  treating  this  part  of  the  subject,  and  striving  to 
keep  in  mind  the  actual  physical  conditions,  by  picturing  the 
armature  conductors  cutting  through  the  flux  lines,  it  is  hoped 
that  the  difficulties  of  the  subject  may,  to  a  great  extent,  be 
removed. 

The  inductance  of  the  windings,  in  so  far  as  it  affects  regu- 
lation, will  be  taken  up  again  in  Chap.  XIV,  and  for  our  present 
purpose — which  is  mainly  to  design  an  armature  that  shall 
not  attain  too  high  a  temperature — it  is  not  proposed  to  add 
much  to  what  was  said  in  Chap.  VIII  when  treating  of  the  flux 
cut  by  the  coil  undergoing  commutation.  A  distinction  was 
then  made  between  the  slot  flux  and  the  end  flux.  The  same 
conditions  are  met  with  in  the  alternator,  where  what  is  usually 
referred  to  as  the  reactive  voltage  component  (the  vector  EgP 
in  Figs.  99  and  100)  is  really  due  to  the  cutting  of  the  end  flux 
by  the  conductors  projecting  beyond  the  ends  of  the  slots:  the 
slot  flux,  being  actually  provided  by  the  main  poles,  does  not 
enter  the  armature  core  below  the  teeth,  and  since  it  is  not  cut 
by  the  armature  inductors,  it  should  not  be  thought  of  as  pro- 
ducing an  e.m.f.  of  self-induction  in  the  windings.  The  slot 


264 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


flux  may  be  considerable,  especially  on  a  heavy  load  of  low  power 
factor,  and  it  will  have  an  appreciable  effect  on  the  inherent 
regulation  of  the  machine.  The  m.m.f.  of  the  conductors 
in  the  slot  accounts  for  the  fact  that  a  certain  percentage  of  the 
flux  in  the  air  gap  does  not  enter  the  armature  core  below  the 
teeth;  but  the  point  here  made  is  that  the  slot  leakage  flux  does 
not  induce  a  counter  e.m.f.  in  the  armature  windings. 

The  core  loss  will  depend  upon  the  flux  necessary  to  develop 
the  e.m.f.  OEg  of  Figs.  99  and  100,  where  the  component  EgP  is  the 
reactance  voltage  drop  due  to  the  flux  linkages  of  the  end  con- 
nections only;  or,  in  other  words,  where  EgP  is  the  e.m.f.  induced 
in  the  conductors  outside  the  slots  by  the  cutting  of  the  flux  lines 
created  by  the  currents  in  all  the  phase  windings. 

87.  Calculation  of  Armature  Inductance. — The  flux  cut  by 
the  conductors  which  project  beyond  the  ends  of  the  armature 
slots  is  very  difficult  to  calculate,  and  empirical  formulas  based 


a 


a 


FIG.  101. — Illustrating  flux  cut  by  end  projections  of  armature  coils. 

on  experimental  data  are  generally  used  for  predicting  the  prob- 
able value  of  the  inductance  of  the  armature  end  connections. 
In  Fig.  101  let  r  represent  the  pole  pitch  of  a  three-phase  generator 
and  V  the  average  axial  extension  of  the  coils  beyond  the  ends 
of  the  armature  core.1  If  the  total  flux  produced  by  the  arma- 
ture currents,  in  the  space  r  cm.  wide  by  V  cm.  deep,  as  shown 
cross-hatched  in  Fig.  101,  can  be  estimated,  the  voltage  developed 
by  the  cutting  of  this  flux  can  readily  be  calculated.  Without 
attempting  to  go  into  the  niceties  of  mathematical  analysis — 
which  would  apply  only  to  one  of  the  many  different  arrange- 
ments of  coils — it  can  be  shown  that  the  flux  produced  through 

1  The  equivalent  projection  of  a  coil  that  is  bent  up  to  clear  the  coils  of 
other  phases  might  be  considered  equal  to  the  projection  of  the  same  coil  if 
flattened  out.  With  r  expressed  in  inches,  an  approximate  value  for  l't 
based  on  the  assumptions  made  in  Art.  83,  is 


=  1.27  (k.v. 


+3 


7     cm. 


ARMATURE  WINDINGS  265 

air  paths  by  the  currents  in  the  axial  prolongations  of  the  slot 
conductors  will  depend  mainly  upon  the  number  of  ampere- 
conductors per  pole  on  the  armature,  and  on  the  amount  of  the 
projection  I'.  There  will  be  no  exact  proportionality  between 
ampere-conductors  per  pole  and  flux,  the  relation  being  a 
logarithmic  function  of  the  pole  pitch  r  and  dependent  on  the 
number  of  slots,  i.e.,  whether  the  winding  is  concentrated  or 
distributed.  The  flux  produced  by  the  connections  running 
approximately  parallel  to  the  circumference  of  the  armature  will 
depend  not  only  on  r  but  also  on  I'.  Thus,  the  amount  of  the 
projection  I'  beyond  the  ends  of  the  slot  would  seem  to  be  a 
more  important  factor  than  the  circumferential  width  of  the 
coils  in  determining  the  end  flux,  and  for  the  calculation  of  the 
total  end  flux  per  pole  (both  ends)  in  the  case  of  a  three-phase 
generator  the  writer  suggests  the  empirical  formula 

*e  =  kTJcle  -     '-=  log*  (12n.l')  (98) 

7l«  -f-  O 

where    Ts  =  the  number  of  inductors  in  each  slot. 
n,  =  the  number  of  slots  per  pole  per  phase. 

V  —  the  projection  of  coil-ends  beyond  end  of  slots,  in 
centimeters. 

le  =  (2r  -f  4Z')  =  approximately  the  total  length  in 
centimeters  per  turn  of  wire  in  a  coil,  less  the  slot 
portion. 

Ic  =  the  armature  current  per  conductor  (r.m.s.  value). 

k  =  constant,  approximately  unity,  depending  upon  the 
design  of  the  machine,  the  arrangement  of  the  wind- 
ings, and  the  proximity  of  masses  of  iron  tending 
to  increase  the  induction. 

The  quantities  6/(n.  +  5)  and  log™  (12n/)  are  factors  in- 
troduced mainly  to  correct  for  the  increase  of  flux  with  a  con- 
centrated winding,  and  for  the  fact  that  the  projection  V  of  the 
coils  will  influence  the  total  flux  to  a  greater  extent  than  the  end 
length  r  which  appears  in  the  expression  for  the  total  length  le. 

If  p  is  the  number  of  poles  of  the  machine,  the  total  number 
of  conductors  per  phase  is  pT8ns,  and  the  average  value  of  the 
voltage  developed  in  the  end  connections  by  the  cutting  of  the 
end  flux  will  be  2f$epTans  X  10~8.  Assuming  the  form  factor 
to  be  1.11,  which  would  be  correct  if  the  flux  distribution  were 
sinusoidal,  and  substituting  for  $e  the  value  given  by  formula 


266          PRINCIPLES  OF  ELECTRICAL  DESIGN 

(98),  the  voltage  component  developed  per  phase  winding  by  the 
cutting  of  the  end  flux  is 

Ee  =  (2.22k)fpTfl.  (^ps)  logio  (12w/)  X  Je  X  10~8     (99) 

This  quantity  is  usually  referred  to  as  the  reactive  voltage 
drop  per  phase  due  to  the  inductance  of  the  end  connections; 
it  appears  as  the  vector  PEg  in  Figs.  99  and  100.  If  the  mul- 
tiplier (2.22k)  be  taken  as  2.4,  the  formula  agrees  well  with 
the  average  of  tests  on  machines  of  normal  design. 

88.  Total  Losses  to  be  Radiated  from  Armature  Core.  — The 
losses  in  the  iron  stampings — teeth  and  core — are  calculated  as 
explained  in  Chap.  VI  (Art.  31).  The  flux  to  be  carried  at  full 
load  by  the  core  below  the  teeth  is  that  which  will  develop  the 
necessary  e.m.f .  as  obtained  from  the  vector  construction  of 
Fig.  100.  The  radial  depth  of  the  armature  stampings  is  cal- 
culated by  assuming  a  reasonable  flux  density  in  the  iron.  This 
will  usually  be  between  7,000  and  8,500  gausses  in  60-cycle 
machines,  increasing  to  10,000  or  even  11,000  in  25-cycle 
generators. 

The  permissible  density  in  the  teeth,  as  previously  mentioned, 
rarely  exceeds  16,000  gausses  at  60  cycles  and  18,000  gausses  at 
25  cycles.  Higher  densities  may  have  to  be  used  occasionally, 
but  special  attention  must  then  be  paid  to  the  methods  of  cool- 
ing, in  order  to  avoid  excessive  temperatures.  The  tooth  density 
being  appreciably  lower  than  in  D.C.  machines,  the  apparent 
flux  density  at  the  middle  of  the  tooth  may  be  used  for  estimating 
the  watts  lost  per  pound.  The  maximum  value  of  the  tooth 
density  will  depend  upon  the  maximum  value  of  the  air-gap 
density,  and  this,  in  turn,  is  modified  by  armature  distortion  and 
slot  leakage.  The  flux  that  must  enter  the  core  and  be  cut  by 
the  armature  inductors  is  known,  but  the  amount  of  flux  enter- 
ing the  teeth  under  each  pole  face  is  greater,  since  it  includes  the 
slot  leakage  flux  in  the  neutral  zone,  the  .amount  of  which  de- 
pends not  only  upon  the  current  in  the  armature,  but  also  upon 
its  phase  displacement,  i.e.,  upon  the  power  factor  of  the  load. 
Then,  again,  the  maximum  value  of  the  air-gap  flux  density  de- 
pends not  only  upon  the  average  density,  but  also  on  the  shape 
of  the  flux  distribution  over  the  pole  pitch.  It  will  not .  be 
necessary  to  go  into  details  of  this  nature  for  the  purpose  of 
estimating  the  temperature  rise  of  the  armature,  and  a  sinusoidal 


ARMATUR^  WINDINGS  267 

flux  distribution  may  be  assumed,  making  the  maximum  air- 

7T 

gap  density  „  times  the  average  value  over  the  pole  pitch.     The 

Zt 

calculation  of  slot  leakage  flux  will  be  explained  later,  and  its 
effect  may  for  the  present  be  neglected.1 

As  a  check  on  the  calculated  core  loss,  the  figures  of  Art.  32 
(page  104)  may  be  used;  but  these  values  will  depend  upon 
whether  the  copper  or  the  iron  losses  are  the  more  important, 
i.e.,  on  the  relative  proportions  of  iron  and  copper  in  the  machine. 
Iron  losses  50  per  cent,  in  excess  of  the  average  values  given  on 
page  104  would  not  necessarily  betoken  inefficiency  or  a  high 
temperature  rise. 

When  computing  the  total  losses  to  be  carried  away  in  the 
form  of  heat  from  the  surface  of  the  armature  core,  the  whole  of 
the  copper  loss  should  not  be  added  to  the  iron  loss,  but  only 
the  portion  of  the  total  PR  loss  which  occurs  in  the  buried  part 
of  the  winding,  i.e.,  in  the  " active"  conductors  of  length  la. 
In  the  case  of  large  machines,  it  may  be  necessary  to  make  some 
allowance  for  eddy-current  loss  in  the  armature  conductors. 
This  loss  might  be  considerable  if  solid  conductors  were  used; 
but  it  is  usual  to  laminate  the  copper  in  the  slot  so  that  the 
eddy-current  loss  due  to  the  slot  flux  is  very  small.  This  point 
must  not,  however,  be  overlooked  in  large  units ; .  and  special 
means  may  have  to  be  adopted  to  avoid  eddy-current  loss  in  the 
armature  conductors. 

89.  Temperature  Rise  of  Armature. — The  probable  tem- 
perature rise  of  the  armature  is  estimated  as  explained  in  Art. 
34  of  Chap  VI,  in  connection  with  the  design  of  D.C.  dynamos. 
The  cooling  surfaces  are  calculated  in  a  similar  manner;  but  with 
the  stationary  armature  and  internal  rotating  field  magnets,  the 
belt  of  active  conductors  is  the  inside  cylindrical  surface  of  the 
armature;  and  this  is  cooled  by  the  air  thrown  against  it  by  the 
fanning  action  of  the  rotor.  The  cooling  coefficient,  contain- 
ing the  factor  v  (the  peripheral  velocity),  may  be  used,  just  as  if 
the  armature  were  rotating  instead  of  the  field  magnets.  The 
external  cylindrical  surface  of  the  armature  core  will  have  no 
air  blown  against  it  (in  the  self -ventilating  machine),  and  the 
value  of  v  in  the  formula  will  be  zero.  In  regard  to  the  radial 

1  The  amount  of  the  slot  leakage  flux,  expressed  as  a  percentage  of  the 
total  flux  per  pole,  becomes  of  importance  in  well-designed  machines  only 
when  the  pole  pitch  is  very  small. 


268          PRINCIPLES  OF  ELECTRICAL  DESIGN 

vent  ducts,  the  cooling  is  not  quite  so  good  as  when  the  armature 
rotates,  but  a  blast  of  air  is  driven  through  the  ducts,  and  this 
is  effective  in  carrying  off  the  heat.  The  difficulty  in  deter- 
mining cooling  coefficients  that  shall  be  applicable  to  all  sizes 
and  types  of  machine  stands  in  the  way  of  obtaining  great 
accuracy  in  the  calculation  of  temperature  rise.  It  is,  however, 
suggested  that  the  formulas  (53)  and  (55)  of  Art.  34  (page  110) 
be  used,  and  that  the  temperature  rise  as  calculated  by  the 
application  of  these  formulas  be  increased  20  per  cent.  A  tem- 
perature rise  of  45°  is  usually  permissible. 

In  designing  steam-turbine-driven  machines  with  forced  venti- 
lation, the  quantity  of  air  required  to  carry  off  the  heat  losses 
must  be  estimated  (see  Art.  34,  page  112)  and  the  size  and  con- 
figuration of  the  various  air  passages  must  be  carefully  studied 
with  a  view  to  preventing  very  high  air  velocities  and  consequent 
increase  of  loss  by  friction.  The  average  velocity  of  the  air 
through  the  ducts  of  machines  provided  with  forced  ventilation 
is  usually  between  1,500  and  4,000  ft.  per  minute.  This  velocity 
should  preferably  not  exceed  5,000  ft.  per  minute;  it  is  usually 
possible  to  keep  within  this  limit  by  carefully  designing  the 
system  of  ventilation. 

High-speed  machines  such  as  turbo-alternators,  when  pro- 
vided with  forced  ventilation,  are  usually  totally  enclosed, 
the  air  passages  being  suitably  arranged  to  prevent  the  out- 
going (hot)  air  being  mixed  with  the  incoming  (cool)  air.  Large 
ducts  must  be  provided  for  conveying  the  air  to  and  from  the 
machine.  A  safe  rule  is  to  provide  ducts  or  pipes  of  such  a 
cross-section  that  the  mean  velocity  of  the  air  will  not  exceed 
2,000  ft.  per  minute. 


CHAPTER  XIII 
AIR-GAP  FLUX  DISTRIBUTION— WAVE  SHAPES 

90.  Shape  of  Pole  Face. — When  an  alternator  is  provided 
with  salient  poles,  the  open-circuit  flux  distribution  over  the 
pole  pitch  can  be  made  to  approximate  to  a  sine  curve  by  suit- 
ably shaping  the  pole  face.  One  method  of  increasing  the  air- 
gap  reluctance  from  the  center  outward  is  illustrated  in  Fig.  102. 
The  " equivalent"  air  gap,  5e,  at  center  of  pole  face  is  calculated 
as  explained  in  Art.  36  of  Chap.  VII  (formula  58),  and  the 


(Electrical) 
About  56° 


cos  or 


FIG.  102. — Method  of  shaping  pole  face  of  salient  pole  alternator, 
equivalent  gap  at  any  other  point  under  the  pole  is  made  equal 
where  a  is  the  angle  (electrical  space  degrees)  between 


to 


cos  a 


the  center  of  pole  and  the  point  considered.  The  pole  face  would 
extend  about  56  degrees  on  each  side  of  the  center,  the  pole  tips 
being  rounded  off  with  a  small  radius.  In  practice  the  curve 
of  the  pole  face  would  probably  not  conform  exactly  with  this 
cosine  law;  it  would  generally  be  a  circular  arc,  not  concentric 
with  the  bore  of  the  armature,  but  with  the  center  displaced  so 

269 


270 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


that  the  air  gap  near  the  pole  tip  would  be  approximately  as 
determined  by  the  method  here  described. 

With  the  cylindrical  field  magnet  it  is  not  usual  to  shape  the 
pole  face.  The  clearance  between  the  tops  of  the  teeth  on 
armature  and  rotor  would  have  a  constant  value,  the  proper 
distribution  of  flux  over  the  armature  surface  being  obtained 
by  spreading  the  field  coils  over  the  periphery  of  the  cylindrical 
rotor.  From  15  to  25  per  cent,  of  the  pole  pitch  is  left  unwound 
at  the  center  of  the  pole.  This  unwound  portion  is  usually 
slotted,  but  it  can  be  left  solid  if  it  is  desired  to  reduce  the  re- 


Alternative  Three-Part 
Wedge,  which  allows  of 
Central  Wedge  being 
preased  down  while  the 
Steel  Liners  are  driven  In 


Detail  of  Slot  and 
Wedge 


FIG.  103. — Rotor  of  four-pole  turbo-alternator,  with  radial  slots. 

luctance  of  the  air  gap  at  the  center  of  the  pole  face  while  yet 
retaining  the  cylindrical  form  of  rotor.  One  advantage  of 
equally  spaced  slots  over  the  surface  of  the  rotor  is  that  there 
are  no  sudden  changes  in  the  air-gap  permeance — the  average 
value  of  which  is  then  the  same  at  all  points  on  the  periphery— 
and  another  argument  in  favor  of  slotting  the  unwound  portion 
of  the  pole  face  is  that  the  field  may  be  " stiffened"  by  using 
high  flux  densities  in  the  teeth. 

Fig.  103  is  a  section  through  part  of  the  rotor  of  a  four- 
pole  turbo-alternator.     The  rotor  of  a  two-pole  machine  may 


AIR-GAP  FLUX  DISTRIBUTION 


271 


be  constructed  in  the  same  way,  with  radial  slots,  but  parallel 
slots  as  shown  in  Fig.  104  are  sometimes  used.  The  calculation 
of  windings  for  the  cylindrical  type  of  field  magnet  will  be  taken 
up  in  Art.  93. 


FIG.  104. — Rotor  of  two-pole  turbo-alternator,  with  parallel  slots. 

91.  Variation  of  Permeance  over  Pole  Pitch — Salient-pole 
Machines. — The  curve  representing  the  variations  of  per- 
meance between  pole  face  and  armature  core  over  the  entire 
pole  pitch,  can  be  drawn  exactly  as  in  the  case  of  the  D.C. 
design.  This  was  explained  in  Arts.  39  and  41  of  Chap.  VII. 
The  pole  shoe  in  Figs.  42  and  43  is  shaped  in  accordance  with  the 
cosine  law  as  explained  in  the  preceding  article,  and  the 
flux  lines  in  Fig.  43  are  therefore  such  as  would  enter  the 
armature  of  a  salient-pole  alternator  on  open  circuit.  The 
practical  construction  of  Fig.  45  has  been  carried  out  on  a  pole 
of  similar  shape,  and  nothing  more  need  be  added  here  concern- 
ing the  manner  of  plotting  a  permeance  curve  similar  to  the  one 
shown  in  Fig.  44. 

The  effect  of  tooth  saturation  may  be  dealt  with  also  as  in  the 
D.C.  design  (Art.  42),  and  a  set  of  curves  such  as  those  of  Fig. 
49  should  be  drawn.  On  account  of  the  higher  frequencies, 
the  tooth  densities  are  lower  in  A.C.  than  in  D.C.  machines, 
and  the  effect  of  saturation  of  the  armature  teeth  is  therefore 
less  noticeable. 


272  PRINCIPLES  OF  ELECTRICAL  DESIGN 

92.  M.m.f.  and  Flux  Distribution  on  Open  Circuit — Salient- 
pole  Machines. — The  procedure  here  is  still  the  same  as  in  D.C. 
design,  and  the  reader  is  referred  to  Arts.  40,  41,  and  42,  of 
Chap.  VII. 

In  deriving  the  curve  of  m.m.f.  from  the  open-circuit  flux 
distribution  curve  (A),  the  modified  method,  as  explained  in 
Chap.  X  under  items  (72)  to  (76),  may  be  used.  This  short 
cut  is  permissible  in  predetermining  the  air-gap  flux  distribution 
of  almost  any  alternating-current  generator,  because,  as  pre- 
viously mentioned,  the  effect  of  low  flux  densities  in  the  teeth 
is  to  discount  their  influence  on  the  distribution  of  the  flux 
density  over  the  armature  surface. 

By  carefully  shaping  the  pole  face,  a  sinusoidal  distribution 
of  flux  density  over  the  armature  surface  can  be  obtained  on 
open  circuit;  but  the  design  of  a  salient-pole  machine  to  give 
a  sine  wave  of  e.m.f.  under  all  conditions  of  loading  involves 
other  factors,  and  is  by  no  means  a  simple  matter.  The  effect 
of  the  armature  m.m.f.  will  be  considered  after  taking  up  the 
special  case  of  the  cylindrical  field  magnet. 

93.  Special  Case  of  Cylindrical  Field  Magnet  with  Distributed 
Winding. — In  the  case  of  a  slotted  rotor  carrying  the  field  coils, 
and  an  air  gap  of  constant  length — due  to  the  fact  that  the  bore 
of  the  stator  or  armature  is  concentric  with  the  (cylindrical) 
rotor — the  shape  of  the  m.m.f.  curve  due  to  field  excitation  alone 
can  readily  be  found  without  resorting  to  the  somewhat  tedious 
process  of  getting  the  permeance  between  pole  and  armature 
points,  as  described  and  recommended  for  salient-pole  machines. 

If  the  whole  surface  of  the  rotor  is  provided  with  equally 
spaced  slots,  the  average  permeance  of  the  air  gap  between 
stator  and  rotor  will  have  a  constant  value  for  all  points  on  the 
armature  periphery.  This  condition  is  represented  in  Fig.  105, 
if  the  center  portion  of  the  pole,  of  width  W,  is  slotted  as  indicated 
by  the  dotted  lines.  This  constant  average  air-gap  permeance 
can  be  calculated  within  a  close  degree  of  approximation  by  mak- 
ing conventional  assumptions  in  regard  to  the  path  of  the  mag- 
netic lines,  as  was  done  in  the  case  of  the  salient-pole  machines 
when  deriving  formula  (57)  (page  117)  giving  the  permeance  at 
center  of  pole.  The  flux  lines  can  be  supposed  to  be  made  up 
of  straight  lines  and  quadrants  of  circles;  and  if  the  permeance 
over  one  tooth  pitch  is  worked  out  for  different  relative  positions 
of  field  and  armature,  very  satisfactory  results  can  be  obtained 


AIR-GAP  FLUX  DISTRIBUTION 


273 


by  this  method.  It  will  usually  suffice  to  make  the  calculations 
for  one  tooth  pitch  in 'the  position  of  greatest  permeance,  and 
again  in  the  position  of  least  permeance.  The  average  of  these 
calculated  values,  divided  by  the  area  of  the  tooth  pitch  in  square 
centimeters,  will  give  the  average  value  of  the  air-gap  permeance 
per  square  centimeter. 

Under  this  condition  of  constant  air-gap  permeance,  the  flux 
distribution  on  open  circuit  will  follow  the  shape  of  the  m.m.f. 
curve;  but,  in  any  case,  since  B  =  m.m.f.  X  permeance  per 
square  centimeter,  the  flux  curve  can  always  be  obtained  when 
the  m.m.f.  distribution  is  known.  Thus,  if  the  portion  W 


M.M.F.  Curve 

for  Field  Winding  only         \  \ 


FIG.  105. — M.M.F.  over  pole  pitch,  due  to  distributed  field  winding. 

of  the  pole  (Fig.  105)  is  not  slotted,  the  permeance  curve,  in- 
stead of  being  a  straight  line  of  which  the  ordinates  are  of  con- 
stant value,  would  be  generally  as  shown  in  Fig.  106.  From 
these  values  of  the  permeance  per  square  centimeter  of  armature 
surface,  curves  such  as  those  of  Fig.  49  (page  133)  can  be  drawn, 
so  as  to  include  the  reluctance  of  the  armature  teeth.  From  all 
points  on  the  armature  between  A  and  B,  and  C  and  D  (Fig. 
106),  the  average  air-gap  permeance  would  have  the  value  A  A'. 
Over  the  central  portion  W  it  would  have  the  value  EE',  while, 
in  the  neighborhood  of  the  points  B  and  C,  it  may  be  assumed 
to  have  an  intermediate  value  as  indicated  by  the  ordinate  BB'. 

18 


274 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


If  the  depth  of  the  slots  in  the  rotor  has  been  decided  upon 
and  the  number  of  ampere-conductors  in  each  slot  determined, 
the  distribution  of  m.m.f.  over  armature  surface  due  to  the 
field  winding  can  readily  be  plotted  as  in  the  lower  sketch  of 
Fig.  105.  Thus  the  ampere-turns  in  the  coil  nearest  the  neutral 
zone  are  represented  by  the  height  AB,  those  in  the  middle  coil 
by  BC,  and  those  in  the  smallest  coil  by  CD.  The  broken  straight 
line  so  obtained  is  best  replaced  by  the  dotted  curve,  which  takes 
care  of  fringing,  and  represents  the  average  effect.  This  curve, 
being  the  open-circuit  distribution  of  m.m.f.  over  armature 
surface  for  a  given  value  of  exciting  current,  may  be  combined 
with  the  curve  of  armature  m.m.f.  to  obtain  the  resultant 
m.m.f.  under  loaded  conditions.  The  flux  curves  for  open- 
circuit  and  loaded  conditions  can  then  be  derived  by  proceed- 
ing exactly  as  in  the  case  of  the  salient-pole  designs. 


A' 


B  E 


D 


FIG.  106. — Distribution  of  air-gap  permeance  in  turbo-alternator 
when  unwound  portion  of  pole  face  is  not  slotted. 

*  With  salient-pole  machines  variations  in  the  open-circuit 
flux  distribution  can  be  obtained  only  by  shaping  the  pole  shoes 
so  as  to  alter  the  air  gap  over  the  pole  pitch  in  a  manner  which 
is  not  easily  determined  except  by  trial,  but  with  the  field  wind- 
ing distributed  in  slots  it  is  not  difficult  to  arrange  the  coils  to 
give  any  desired  distribution  of  m.m.f.  over  the  pole  pitch. 
Thus,  if  the  desired  flux  curve  is  known  (a  sinusoidal  distribu- 
tion is  usually  best  for  alternating-current  machines),  and  if  the 
average  permeance  per  square  centimeter  of  the  air  gap  has  been 
calculated,  an  ideal  m.m.f.  curve  can  easily  be  drawn,  since,  at 
every  point,  the  m.m.f.  is  the  ratio  of  the  flux  density  to  the 
permeance  per  square  centimeter.  The  spacing  and  depth  of 
slots  can  then  be  arranged  to  produce  a  magnetizing  effect  as 
nearly  as  possible  the  same  as  that  of  the  ideal  curve. 

94.  Armature  M.M.F.  in  Alternating-current  Generators. — 
In  a  continuous-current  machine  the  current  has  the  same  value 


AIR-GAP  FLUX  DISTRIBUTION  275 

at  all  times  in  all  the  armature  conductors,  and  equation  65 
(page  136)  shows  how  the  armature  m.m.f .  follows  a  straight-line 
law  over  a  zone  equal  to  the  pole  pitch,  this  being  the  distance 
between  brushes  referred  to  armature  surface.  In  alternating- 
current  and  polyphase  generators  the  curve  of  armature  m.m.f. 
can  no  longer  be  represented  graphically  by  straight  lines  as  in 
Fig.  53,  because  the  value  of  the  current  will  not  be  the  same  in 
all  the  conductors  included  in  the  space  of  a  pole  pitch. 

Considering  first  the  polyphase  synchronous  generator,  and 
assuming  a  sinusoidal  current  wave,  it  is  an  easy  matter  to  draw 
a  curve  representing  the  armature  m.m.f.  at  any  particular  in- 
stant of  time,  provided  the  phase  displacement — or  position  of 
the  conductors  carrying  the  maximum  current — relatively  to 
center  line  of  pole  is  known.  If  this  be  done  for  different  time 
values,  a  number  of  curves  will  be  obtained,  all  consisting  of 
straight  lines  of  varying  slopes,  the  length  of  which  relatively 
to  the  pole  pitch  will  depend  on  the  number  of  phases  for  which 
the  machine  is  wound.  The  average  of  all  these  curves  will  be 
a  sine  curve  of  which  the  position  in  space  relatively  to  the 
poles  is  constant,  and  exactly  90  electrical  space  degrees  behind 
the  position  of  maximum  current. 

The  method  of  drawing  the  curve  of  armature  m.m.f.  for  any 
instant  of  time,  is  illustrated  in  Fig.  107,  where  the  upper  diagram 
shows  the  distribution  of  m.m.f.  over  the  armature  periphery 
of  a  three-phase  generator  at  the  instant  when  the  current  in 
phase  (2)  has  reached  its  maximum  value.  If  the  power  factor 
is  unity  (load  non-inductive),  the  current  maximum  will  occur 
simultaneously  with  the  voltage  maximum,  i.e.,  when  the  belt  of 
conductors  is  under  the  center  of  the  pole  face,  as  shown  in  the 
diagram.  A  low  power  factor  would  cause  the  current  to  attain 
its  maximum  value  only  after  the  center  of  the  pole  has  travelled 
an  appreciable  distance  beyond  the  center  of  the  belt  of  conduc- 
tors, and  this  effect  will  be  explained  later;  at  present  we  are 
concerned  merely  with  the  distribution,  and  magnitude,  of  the 
armature  m.m.f.  The  vector  diagram  on  the  right-hand  side 
of  the  (upper)  figure  shows  how*  the  value  of  the  current  in 
phases  (1)  and  (3),  at  the  instant  considered,  will  be  exactly  half 
the  maximum  value;  and  the  magnetizing  effect  of  phase  (1) 
or  (3)  is  therefore  exactly  half  that  of  phase  (2).  The  angle  of 
60  degrees  between  vectors  representing  three-phase  currents — 
with  a  phase  displacement  at  terminals  of  120  degrees — is 


276 


PRINCIPLES  OF"  ELECTRICAL  DESIGN 


accounted  for  by  the  fact  that  the  angular  displacement  between 

180 
adjoining  belts  of  conductors  is  only  -^—  —  60  electrical  degrees; 

o 

but  the  connections  between  the  phase  windings  are  so  made  as 
to  obtain  the  phase  difference  of  120  degrees  between  the  re- 
spective e.m.fs.  Thus,  the  reversal  of  the  vector  (1)  in  the 


_ 180° 

loU  5^1 


Direction  of  Travel  of  Poles 


Current=  0 

V 


(2)  |  (3) 

I 
Current  !=0.866  /max.  ! 


/  cos  30  =  0.866  / 


FIG.  107. — Instantaneous  values  of  armature  m.m.f.  in  three-phase 
v  generator. 

diagram  would  cause  it  to  lead  vector  (2)  by  120  degrees  instead 
of  lagging  behind  by  60  degrees;  and  the  reversal  of  the  vector  (3) 
would  cause  it  to  be  120  degrees  behind  (2)  instead  of  60  degrees 
ahead. 

The  lower  diagram  of  Fig.  107  shows  the  armature  m.m.f. 
one-twelfth  of  a  period  later,  i.e.,  when  the  poles  have  moved  to 
the  left  (or  the  conductors  to  the  right)  30  electrical  space  de- 


AIR-GAP  FLUX  DISTRIBUTION 


277 


grees.  The  current  in  phases  (1)  and  (2)  now  has  the  instan- 
taneous value  i  =  Imax  cos  30  =  0.866/max;  while  the  current 
in  (3)  is  zero.  If  several  curves  of  this  kind  are  drawn,  it  will 
be  found  that  the  instantaneous  values  of  m.m.f .  at  any  point  on 
the  armature  periphery  (considered  relatively  to  the  poles)  differs 
very  little  from  the  average  value;  in  other  words,  the  pulsations 
of  flux  due  to  cyclic  changes  in  the  m.m.f.  will,  in  a  three-phase 
machine,  be  negligibly  small.  For  this  reason,  and  also  in  order 
to  shorten  and  simplify  the  work,  the  armature  m.m.f.  of  a 
polyphase  generator  may  conveniently  be  studied  by  assuming  a 
large  number  of  conductors,  and  a  number  of  phases  equal  to 


Lag  of  Current  Behind  Open  Circuit  E.M.F. 
Corresponding  to  Brush  Shift  in  D.C.  Machines 

FIG.   108. — Method  of  obtaining  armature  m.m.f.  curve  from  curve  of 
current  distribution. 


the  number  of  conductors  in  the  space  of  one  pole  pitch.  Thus, 
the  ordinates  of  curve  I  of  Fig.  108  (assumed  to  be  a  sine  curve) 
give  the  value  of  the  current  in  the  various  conductors  distributed 
over  the  armature  surface.  It  is  understood  that  the  current 
in  each  individual  conductor  varies  according  to  the  simple 
harmonic  law;  but  it  is  constant  in  value  for  a  given  position  on 
the  armature  surface  considered  relatively  to  the  poles.  The 
direction  of  the  current  in  the  conductors  between  the  points 
A  and  B  may  be  considered  as  being  downward,  while  the  direc- 
tion of  the  current  in  the  adjoining  section  of  width  T  would  be 
upward.  The  maximum  value  of  the  armature  m.m.f.,  there- 
fore, occurs  at  the  point  B,  and  we  may  write: 


278          PRINCIPLES  OF  ELECTRICAL  DESIGN 

Maximum  value  of  armature  ampere-turns  per  pole  =  average 
value  of  current  in  section  OB  X  number  of  turns,  or 

(SI)a    =*ImaX    X   ~  (100) 

where  Z'  stands  for  the  total  number  of  inductors  on  the  arma- 
ture periphery.     This  fnay  be  compared  directly  with  formula 
66   (page   136),   which  applies  to  direct-current  machines,   by 
putting  it  in  the  form 
Max.  m.m.f.  (gilberts)  per  pole 

0.47T  X  2  X  Ic  V2Z' 


1.11  X2p 


where  Ic  stands  for  the  virtual  or  r.m.s.  value  of  the  current  in 
the  armature  conductors. 

That  the  armature  m.m.f.  curve  in  Fig.  108  is  also  a  sine  curve 
when  the  current  follows  the  sine  law  is  easily  seen  from  the 
general  solution,  thus: 

The  magnetizing  effect  of  the  conductors  in  the  small  space  of 
width  dd  is 

zr  de 

Imax  sin  0  X  — 

V     » 

rrt      in 

wherein  lmax  sin  6  is  the  current  per  conductor,   and   —  —  is 

the  number  of  conductors  in  the  space  considered  (the  angle  0 
being  expressed  in  radians)  .  • 

The  expression  for  the  total  ampere-conductors  is  therefore 

7i' 

Imax  —  S  sin  6dd 
pw 

With  an  increase  in  the  number  of  inductors  (and  phases)  this 
quantity  approaches  more  and  more  nearly  the  definite  integral, 
i.e.,  the  area  of  the  current  curve,  as  indicated  by  CC'  in  Fig.  108 
being  a  measure  of  the  shaded  portion  of  the  curve  /;  and  we 
can  then  write 

Zf 

Armature  ampere-conductors  =  Imax  ""*/*  sin  0  d6 

Tip 

Zi' 

=    -   Imax  —    COS  0  +  C 


AIR-GAP  FLUX  DISTRIBUTION  279 

the  maximum  value  of  which  occurs  when  6  =  o  and  6  =  IT. 
The  constant  of  integration  merely  determines  the  position  of 
the  datum  line;  and  since  we  have  symmetry  and  equal  strength 
of  North  and  South  poles,  we  can  put  C  =  o  and  write  for  the 
maximum  value  of  the  armature  ampere-turns  per  pole, 

7'  -\/^,7' 

(SI) a  =    -  Imax  =  —  -Ie  which  checks  with  formula  (100). 
wp  irp 

The  angle  of  displacement  0  (Fig.  108)  of  this  curve  relatively 
to  the  center  line  of  pole  depends  upon  the  " internal"  power 
factor,  and  also  upon  the  displacement  of  the  wave  of  developed 
e.m.f.,  a  displacement  or  distortion  which  is  due  to  cross-magne- 
tization. The  angle  0  is  not  very  easily  predetermined,  but, 
once  known  or  assumed,  the  curve  M  can  be  drawn  in  the  correct 
position  relatively  to  the  curve  of  field  m.m.f.;  and  the  resultant 
m.m.f.  over  the  armature  surface  can  be  obtained  exactly  as  for 
the  direct-current  machine  (see  Fig.  53,  page  137).  An  approxi- 
mate method  of  predetermining  the  displacement  angle  /3  for 
any  load  and  power  factor  will  be  explained  in  Art.  98. 

Armature  M.M.F.  Curve  of  Single-phase  Alternator. — When 
single-phase  currents  are  taken  from  an  armature  winding,  the 
m.m.f.  due  to  this  winding  as  a  whole  must  necessarily  be  of  zero 
value  at  the  instant  of  time  when  the  current  is  changing  from  its 
positive  to  its  negative  direction.  This  suggests  that  the  mag- 
netizing effect  of  the  loaded  armature  will  be  pulsating;  that  is 
to  say,  it  cannot  be  of  constant  strength  at  any  given  point  con- 
sidered relatively  to  the  poles,  whatever  may  be  the  phase  dis- 
placement of  the  current  relatively  to  the  developed  voltage.  If 
the  change  of  current  in  any  given  conductor  be  considered  over 
a  complete  cycle,  and  if  at  the  same  time  the  position  of  this  con- 
ductor relatively  to  the  poles  be  noted,  it  will  be  seen  that,  rela- 
tively to  the  field  magnet  system,  the  armature  windings  produce 
a  pulsating  field  of  double  the  normal  frequency.  The  actual 
flux  component  due  to  the  armature  currents  will  not,  however, 
pulsate  to  any  appreciable  extent,  because  the  tendency  to  vary 
in  strength  at  comparatively  high  frequencies  is  checked  by  the 
dampening  effect  of  the  field  coils,  even  if  the  pole  shoes  and 
poles  are  laminated. 

No  modern  single-phase  alternator,  unless  of  very  small  size, 
should  be  built  without  amortisseur  windings,  or  damping  grids. 
These  consist  of  copper  conductors  in  holes  or  slots,  running 
parallel  to  the  shaft,  in  the  faces  of  the  field  poles.  They  are 


280 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


joined  together  at  both  ends  by  heavy  copper  connections,  and 
form  a  " squirrel  cage"  of  short-circuited  bars  which  damp  out 
the  flux  pulsations,  and  also  prevent  the  sweeping  back  and  forth, 
or  "swinging,"  of  the  armature  flux  due  to  "hunting"  when  syn- 
chronous alternating-current  machines  are  coupled  in  parallel. 

Returning  to  the  magnetizing  effect  of  the  single-phase  arma- 
ture, it  is,  therefore,  the  average  or  resultant  armature  m.m.f. 
considered  relatively  to  the  poles  with  which  we  are  mainly  con- 
cerned. The  most  satisfactory  way  of  studying  an  effect  of  this 
kind  is  to  draw  the  actual  m.m.f.  curves  at  definite  intervals  of 


FIG.  109. — Instantaneous  and  average  values  of  armature  m.m.f.  in  single- 
phase  alternator. 

time,  and  then  average  the  values  so  obtained  for  different  points 
on  the  armature  surface,  the  position  of  these  points  being 
considered  fixed  relatively  to  the  field  poles. 

This  has  been  done  in  Fig.  109,  where  the  windings  are  shown 
covering  60  per  cent,  of  the  armature  surface,  and  the  distance  r 
is  one  pole  pitch.  This  distance  is  divided  into  10  equal  parts, 
each  corresponding  to  18  electrical  degrees.  The  thick  line  repre- 
sents the  armature  m.m.f.  when  the  current  has  reached  its 
maximum  value.  The  armature  is  then  supposed  to  move  18 
degrees  to  the  right  of  this  position,  and  a  second  m.m.f.  curve  is 


AIR-GAP  FLUX  DISTRIBUTION  281 

drawn,  corresponding  to  this  position  of  the  windings.  Its 
maximum  ordinate  is,  of  course,  less  than  in  the  case  of  the  first 
curve,  because  the  current  (which  is  supposed  to  follow  the  sine 
law)  now  has  a  smaller  value.  This  process  is  repeated  for  the 
other  positions  of  the  coil  throughout  a  complete  cycle,  and  the 
resultant  m.m.f.  for  any  point  in  space  (i.e.,  relatively  to  the 
poles,  considered  stationary)  is  found  by  averaging  the  ordinates 
of  the  various  m.m.f.  curves  at  the  point  considered.  In  this 
manner  the  curve  M  of  Fig.  109  is  obtained.  It  is  seen  to 
be  a  sine  .curve,  of  which  the  maximum  ordinate  is  half  the 
instantaneous  maximum  m.m.f.  per  pole  of  the  single-phase 
winding,  and  it  may  be  used  exactly  in  the  same  way  as  the 
curve  M  in  Fig.  108  (representing  armature  m.m.f.  of  a  poly- 
phase machine) ;  that  is  to  say,  it  can  be  combined  with  the  field 
pole  m.m.f.  curve  to  obtain  the  resultant  m.m.f.  at  armature 
surface  from  which  can  be  derived  the  flux  distribution  curves 
under  loaded  conditions. 

The  maximum  value  of  the  resultant  ampere-turns  per  pole  is, 
therefore, 

*/».,  X  jp  =  Imax  X    ^  (102) 

where  Z  is  the  total  number  of  armature  face  conductors.  Ex- 
pressed in  gilberts  the  formula  is, 

Maximum  ordinate  of  armature  m.m.f.  }       QAw\/'2lcZ 


curve  in  single-phase  alternator.  2  X  2p 

0.4rrZ/c 


which,  together  with  formula  (102),  may  be  compared  with  the 
formula  (100)  and  (101)  for  polyphase  generators. 

95.  Slot  Leakage  Flux.  —  Referrring  again  to  Fig.  108,  if  we 
wish  to  derive  a  curve  of  resultant  m.m.f.  over  the  armature 
periphery  for  any  condition  of  loading,  it  will  be  necessary, 
before  combining  the  curves  of  armature  and  field  pole  m.m.f., 
to  determine  the  relative  positions  of  these  two  curves.  In  the 
direct-current  machine,  the  position  of  maximum  armature 
m.m.f.  coincides  with  the  brush  position;  but  the  point  B  in 
Fig.  108  is  not  so  easily  determined.  Its  distance  from  the 
center  of  the  pole  is  0  +  90°,  a  displacement  which  depends 
not  only  on  the  power  factor  of  the  load  (i.e.,  on  the  lag  of  the 
current  behind  the  terminal  potential  difference),  but  also  on  the 
strength  of  the  field  relatively  to  the  armature,  because  this 


282 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


relation  determines  the  position  (relatively  to  the  center  of  the 
pole)  of  the  maximum  e.m.f.  developed  in  the  conductors. 

The  field  m.m.f.  will  depend  upon  the  flux  in  the  air  gap,  and 
since  this  includes  the  slot  leakage  flux,  it  will  be  necessary  to 
consider  the  meaning,  and  determine  the  value,  of  the  slot  flux 
before  attempting  to  calculate  the  angle  0  of  Fig.  108. 

Apart  from  the  action  of  the  armature  winding  as  a  whole, 
causing  a  reduction  of  the  total  flux  crossing  the  air  gap  from 
pole  face  to  armature  teeth,  the  current  in  the  individual  con- 
ductors, by  producing  a  leakage  of  flux  in  the  slots  themselves, 
still  further  reduces  the  useful  flux  when  the  machine  is  loaded. 
The  whole  of  the  flux  entering  the  tops  of  the  teeth  is  not  cut 


j 

''i!        >','!•     i:1,',    :    :!i    !!!!«!  ;  taiill     1!«i    ;.'; :     '•<,  .     $m 

nW  hrr  W^  Mtff,  $M  ,w-  /$w  tm ,  ltHJ--  RM 

T-^K/,//,    vv^fpu«uuf«4uffla^    \\>::^ 


Tti: 


11 
li 


^- 

FIG.  110. — Flux  entering  armature  of  A.C.  generator  under  open  circuit 

conditions. 

by  the  conductors  buried  in  the  slots,  and  the  voltage  actually 
developed  in  the  "  active "  portion  of  the  armature  windings  will 
be  reduced  in  proportion  to  the  amount  of  flux  which,  instead  of 
entering  the  armature  core,  is  diverted  from  tooth  to  tooth. 
This  loss  of  voltage  is  usually  attributed  to  the  reactance  of  the 
embedded  portion  of  the  windings,  and  is  referred  to  as  a  react- 
ance voltage.  This  term,  however,  although  very  convenient,  is 
liable  to  lead  to  confusion  when  an  attempt  is  made  to  realize 
the  physical  meaning  of  armature  reactance.  It  suggests  that  a 
certain  electromotive  force  is  generated  in  the  conductors,  thus 
causing  a  flow  of  current  which,  in  turn,  produces  the  flux  of 
self-induction  and  a  reactive  electromotive  force.  This  is 
incorrect  and  leads  to  a  mistaken  estimate  of  the  actual  amount  of 


AIR-GAP  FLUX  DISTRIBUTION 


283 


flux  in  the  armature  core — a  mistake  of  little  practical  import 
yet  tending  to  obscure  the  issue  when  considering  the  problem 
of  regulation,  and  standing  in  the  way  of  a  clear  conception  of 
the  flux  distribution  in  the  air  gap. 

The  effect  of  the  current  in  the  buried  conductors  will  be  under- 
stood by  comparing  Figs.  110  and  111,  where  the  dotted  lines 
indicate  roughly  the  paths  taken  by  the  magnetic  flux  under 
open-circuit  conditions  (Fig.  110)  and  under  load  conditions 
(Fig.  111).  In  the  first  case,  when  no  current  flows  in  the  arma- 
ture conductors,  the  whole  of  the  flux  entering  the  tops  of  the 
teeth  passes  into  the  armature  core  and  is  cut  by  all  the  conduc- 
tors. In  the  second  case  the  magnetomotive  force  due  to  the 


Direction  of  travel  of  poles 

FIG.  111. — Flux  entering  armature  of  A.C.  generator  when  the  conductors 
are  carrying  current. 

armature  current  diverts  a  certain  amount  of  flux  from  tooth  to 
tooth,  which  since  it  does  not  enter  the  armature  core  is  not  cut 
by  all  the  conductors.  This  conception  of  the  slot  flux,  as  that 
portion  of  the  total  flux  leaving  the  pole  shoe,  which  crosses  the 
air  gap  but  does  not  enter  the  armature  core  below  the  teeth, 
disposes  of  the  difficulties  encountered  by  many  engineers  when 
faced  with  the  necessity  of  calculating  the  slot  inductance.  It 
is  unnecessary  to  consider  the  leakage  flux  in  the  slots  under  the 
pole  face,  but  it  is  important  to  know  the  amount  of  flux  in  the 
neutral  zone, *  which  passes  from  tooth  to  tooth  and  generates  no 

lBy  neutral  zone  is  meant  the  space  between  poles  on  the  armature 
surface  where  the  lines  of  magnetic  flux  are  parallel  to  the  direction  of  travel 
of  the  conductors. 


284 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


electromotive  force  in  the  conductors.  If  this  leakage  slot  flux 
(in  the  neutral  zone)  were  actually  cut  by  the  conductors,  it 
would  generate  a  component  of  electromotive  force  lagging  one- 
quarter  period  behind  the  main  component  (on  the  assumption  of 
sine-wave  form),  and  it  can  therefore  conveniently  be  represented 
in  vector  diagrams  as  if  it  were  an  electromotive  force  of  self- 
induction.  The  quantitative  calculation  of  this  electromotive 
force  will  be  considered  in  Art.  97.  Although  only  brief 
mention  has  been  made  of  the  leakage  flux  in  slots  other  than 
those  in  the  neutral  zone,  it  is  not  suggested  that  this  flux  is 
negligible  in  amount;  but  the  flux  distribution  under  the  pole 
face,  whatever  may  be  its  distortion,  affects  only  the  wave  shape 
(and  form  factor)  of  the  generated  electromotive  force,  and  in  no 
way  alters  the  average  value  of  the  developed  voltage.  The 
difference  between  the  total  flux  entering  the  armature  teeth 
from  each  pole  face  and  the  amount  of  the  slot  flux  (or  the 
equivalent  slot  flux)  in  the  neutral  zone  represents  the  flux 
actually  cut  by  all  the  conductors  on  the  armature. 

96.  Calculation  of  Slot  Leakage  Flux. — The  effect  of  slot  in- 
ductance being  generally  to  reduce1  the  amount  of  the  total  air- 


'  ''' 


<A>  (B) 

FIG.  112.— Flux  entering  teeth  in  neutral  zone;  showing  effect  of  armature 
current  in  producing  slot  leakage. 

gap  flux  which  is  actually  cut  by  the  conductors,  the  simplest 
way  to  obtain  a  quantitative  value  for  the  slot  reactance  is  to 
calculate  the  total  flux  which  leaks  from  tooth  to  tooth  in  the 
neutral  zone.  Diagrams  (A)  and  (B)  in  Fig.  112  indicate  the 
approximate  paths  of  the  magnetic  lines  in  the  neutral  zone,  (A) 
when  the  current  in  the  slot  conductors  is  zero,  and  (B)  when  it 
has  an  appreciable  value.  The  amount  of  flux  diverted  from 
the  armature  core  into  the  leakage  paths  referred  to  may  be 

1  Except  in  the  case  of  a  condenser  load  and  leading  current,  in  which 
case  the  tendency  would  be  to  increase  the  total  useful  flux. 


AIR-GAP  FLUX  DISTRIBUTION 


285 


calculated  by  assuming  the  current  in  the  slot  conductors  to  be 
acting  independently  of  the  field  magnetomotive  force.  Thus,  in 
Fig.  113  the  total  slot  flux  is  the  sum  of  three  component 
fluxes:  3>r  passing  through  the  space  occupied  by  the  copper,  a 
portion  of  which  will  be  cut  by  some  of  the  conductors;  <J>2 
crossing  the  space  above  the  windings,  usually  occupied  by  the 
wedge;  and  <J>3  which  leaks  from  tooth  top  to  tooth  top.  If  the 
conductors  were  concentrated  as  a  thin  layer  at  the  bottom  of 
the  slot,  the  loss  of  voltage  due  to  reduction  of  core  flux  (see 
Fig.  112)  could  be  calculated  by  assuming  that  the  useful  flux  is 
reduced  by  an  amount  equal 

to  the  slot  flux.     The  portion  ,--*'*          ""*"*'*.. 

$1,  however,  in  Fig.  113,  be- 
ing cut  by  some  of  the  con- 
ductors, requires  the  calcu- 
lations to  be  based  on  an 
equivalent  slot  flux  which,  if 
cut  by  all  the  conductors, 
would  develop  an  electromo-  — 


FIG. 


113. — Illustrating  method  of  cal- 
culating slot  leakage  flux. 


tive  force  equal  to  the  actual 

loss    of  pressure.     This   flux 

may  be  calculated  as  follows: 

The  amount  of  flux  in  the  small  strip  dx  deep  (Fig.  113)  of  1  cm. 

axial  length,  i.e.,  perpendicularly  to  the  plane  of  the  paper,  is  d$t 

=  m.m.f .  X  dP,  where  dP  is  the  permeance  of  the  air  path — the 

reluctance  of  any  iron  in  the  path  of  the  lines  being  neglected. 

Whence 

(**!  =  (0.4ir!r.J.)Jxy 

where  T8  is  the  number  of  conductors  per  slot;  7,  is  the  current 
per  conductor  in  amperes;  and  the  dimensions  dl  and  s  (see  Fig. 
113)  are  in  centimeters.  Since,  however,  this  flux  element  (see 
Fig.  112,  B)  is  cut  by  T8  (dl  -  x)/dl  conductors,  the  loss  of 
pressure  is  due  to  the  fact  that  it  is  not  cut  by  Tsxldv  conductors. 
The  " equivalent"  flux  to  cause  the  same  loss  of  pressure  would, 
if  it  did  not  link  with  any  of  the  conductors,  therefore  be 


(equivalent)     ~~    C^Pj    X     ? 


Thus 


(equivalent) 


QAirTJ,  fdl 
~d?T~)0   X*dx 


0.4  ird, 
3s 


TJ8 


286  PRINCIPLES  OF  ELECTRICAL  DESIGN 

The  permeances  of  the  air  paths  of  the  component  fluxes 
$2  and  $3  can  be  calculated  fairly  accurately  (See  Art.  5,  Chap. 
II).  Let  them  beP2  and  P3  respectively.  Then,  if  la  is  the  axial 
length  of  the  armature  core  in  centimeters,  the  total  "  equivalent " 
slot  flux  in  the  neutral  zone  is 

(104) 

97.  Effect  of  Slot  Leakage  on  Full -load  Air-gap  Flux. — Before 
considering  a  method  of  drawing  the  curve  of  air-gap  flux  dis- 
tribution under  load,  it  will  be  advisable  to  determine  what 
should  be  the  area  of  this  curve.  The  area  of  the  required  flux 
distribution  curve  is  a  measure  of  the  total  flux  per  pole  in  the  air 
gap,  and  it  would  be  possible  to  express  this  in  terms  of  the  open- 
circuit  flux  distribution  curve  if  we  knew  the  e.m.f.  that  would 
have  to  be  developed  in  the  armature  windings  on  the  assumption 
of  all  the  flux  passing  from  pole  face  to  armature  teeth  being 
actually  cut  by  all  the  conductors.  It  is  therefore  proposed  to 
determine  what  may  be  called  the  " apparent"  developed  e.m.f., 
that  is  to  say,  the  e.m.f.  that  would  be  developed  in  the  armature 
windings  under  load  conditions  if  it  were  not  for  the  fact  that 
some  of  the  flux  in  the  air  gap  leaks  across  from  tooth  to  tooth  in 
the  neutral  zone,  and  is  not  actually  cut  by  the  conductors. 

Consider  first  the  condition  of  zero  power  factor.  The  current 
then  lags  90  degrees  behind  the  e.m.f.,  and  reaches  its  maximum 
value  in  the  conductors  situated  midway  between  poles.  The 
slot  leakage  flux,  and  the  demagnetizing  effect  of  the  armature 
winding,  will  then  both  have  reached  their  maximum  value. 

In  the  vector  diagram,  Fig.  1 14,  let  OEt  represent  the  required 
terminal  voltage  if  the  machine  is  mesh-connected,  or  the  cor- 
responding potential  difference  per  phase  winding  if  the  machine 
is  star-connected.  (In  a  three-phase  Y-connected  generator  OEt 

would  be  — /-=  times  the  terminal  voltage.)     The  vector  01,  drawn 

V  3 

90  degrees  behind  OEt,  is  the  armature  current  on  zero  power 
factor.  The  impedance-drop  triangle  is  constructed  by  drawing 
EtP  parallel  to  01  of  such  a  length  as  to  represent  the  IR  drop 
per  phase,  and  PEg  at  right  angles  to  01  to  represent  the  reactive 
pressure  drop  per  phase  in  the  end  connections.  OEg  is  therefore 
the  voltage  actually  developed  in  the  slot  conductors,  because  it 
contains  the  component  PEg  to  balance  the  voltage  generated  by 


AIR-GAP  FLUX  DISTRIBUTION  287 

the  cutting  of  the  end  flux,  and  the  component  EtP  to  overcome 
the  ohmic  resistance  of  the  windings.  Now  produce  PEg  to  Ear 
so  that  EgE'g  represents  the  voltage  that  would  be  developed  by 
the  slot  flux  if  this  were  cut  by  the  conductors.  OE'g,  which 
may  be  called  the  apparent  developed  voltage,  is  then  the  elec- 
tromotive force  that  would  have  been  developed  in  the  armature 
windings  if  the  slot  flux  had  actually  entered  the  core  instead  of 
being  diverted  from  tooth  to  tooth  by  the  action  of  the  current 
in  the  conductors.  It  is  therefore  also  a  measure  of  the  total 
flux  passing  through  the  air  gap  into  the  armature  teeth,  and  the 
magnetizing  ampere-turns  necessary  to  produce  this  flux  would, 
on  open  circuit,  actually  develop  this  electromotive  force  in  the 
armature.  Thus  when  the  resultant  magnetomotive  force  in  the 
magnetic  circuit  is  such  that  E'a  volts  would  be  developed  on 
open  circuit,  the  terminal  voltage  under  the  assumed  load  con- 


(5lot£) 


I 

FIG.  114. — Vector  diagram  of  alternator  operating  at  zero  power  factor. 

ditions  would  be  Et.  It  is  usual,  when  the  power  factor  is  zero, 
to  consider  this  loss  of  pressure  as  equal  to  the  total  reactive 
drop  (E'gP]  because,  owing  to  the  relative  smallness  of  PEt  and 
the  fact  that  its  direction  is  such  as  to  have  little  effect  on 
the  pressure  drop,  the  error  introduced  by  this  assumption  is 
negligible. 

The  length  of  the  vector  E0E'g  in  Fig.  114  can  be  calculated 
from  the  known  slot  flux  $es  as  given  by  formula  (104)  of  the 
preceding  article.  If  <i>a  is  the'  flux  per  pole  actually  cut  by  the 
conductors,  the  total  flux  per  pole  in  the  air  gap  under  load  con- 
ditions will  be  <i>  =  3>a  +  2<£es. 

This  total  flux,  if  actually  cut  by  the  armature  conductors, 
would  generate  the  electromotive  force  referred  to  as  the  "ap- 
parent" developed  voltage,  and  represented  by  OE'a  in  Fig.  114. 

The  flux  2$€8  maxwells  is  the  portion  of  the  total  air-gap  flux 
which,  under  load  conditions,  is  no  longer  cut  by  the  armature 


288  PRINCIPLES  OF  ELECTRICAL  DESIGN 


conductors.     The  average  value  of  the  voltage  lost  per  phase 
winding  is  therefore 

2  $    T)N 

&  (average)    =    1  Q8  \X  QC\    '•      snsP) 

or,  since  Np  =  120/ 


E(average) 


Assuming  the  sinusoidal  wave  shape,  it  is  necessary  to  multiply 
by        ,-  to  obtain  the  r.m.s.  value.     Thus 


M.. 

V2X10- 

The  slot  flux  in  the  neutral  zone  will  be  a  maximum  on  zero 
power  factor  when  the  current  Is  producing  it  is  approximately 
equal  to  the  maximum  value  of  the  armature  current,  or  to  \/2lc> 
Inserting  this  value  of  Is  in  formula  (104)  and  substituting  in 
formula  (105)  we  get 

E8  =  2irf  X  OA7rTsznsplaIc~  -f  P2  +  P3   X  10~8    (106) 


This  quantity  is  usually  referred  to  as  the  reactive  voltage  drop 
per  phase  due  to  the  slot  inductance.  It  appears  as  the  vector 
E'aEg  in  Fig.  114.  . 

If  it  were  permissible  to  assume  the  alternating  quantities 
and  the  flux  distribution  in  the  air  gap  to  be  sinusoidal,  the  con- 
struction of  Fig.  114  might  be  repeated.for  conditions  other  than 
zero  power  factor.  These  assumptions  involve  the  idea  of  a 
slot  leakage  flux  diminishing  with  increasing  power  factor,  the 
actual  change  with  varying  angle  of  lag  being  in  accordance  with 
the  sine  law.  This  does  not  take  into  account  tooth  saturation 
and  distortion  of  the  current  wave;  but  as  a  practical  and  ap- 
proximate method  it  is  permissible.  The  vector  diagram  for  any 
power-factor  angle  0  is  then  as  shown  in  Fig.  115.  Here  \I/  is 
the  angle  of  lag  between  the  current  and  the  e.m.f.  actually 
developed  in  the  armature  conductors;  and  cos  ^  is  the  "  internal" 
power  factor.  The  angle  \I/'  shows  the  lag  of  the  current  behind 
the  "apparent"  developed  voltage,  OE'g,  and  it  will  be  seen  that 
the  combined  effect  of  end  flux  and  slot  flux  is  to  reduce  this 
voltage  by  an  amount  approximately  equal  to  PE'g  sin  \l/'. 

98.  Method  of  Determining  Position  of  Armature  M.m.f.  — 
Turning  again  to  Fig.  108  (page  277),  we  are  still  unable  to  de- 
termine the  angle  0,  or  the  displacement  (@  +  90°)  of  the 


AIR-GAP  FLUX  DISTRIBUTION  289 

maximum  armature  m.m.f .  beyond  the  center  line  of  the  pole,  be- 
cause the  angle  \j/f  of  Fig.  115  shows  merely  the  lag  of  the  current 
behind  the  apparent  developed  e.m.f.;  but,  owing  to  armature  dis- 
tortion, the  full-load  flux  distribution  curve  (from  which  the  voltage 
OE'g  is  derived)  will  not  be  symmetrically  placed  relatively  to  the 
center  line  of  the  pole;  it  will  be  displaced  in  the  direction  of 
motion  of  the  conductors,  i.e.,  to  the  right  in  Fig.  108.  With  the 
aid  of  the  vector  diagram  Fig.  115  we  can,  however,  obtain  a 
value  for  the  angle  /3  of  Fig.  108  which  will  enable  us  to  place 
the  curve  of  armature  m.m.f.  in  a  position  relatively  to  the  field 
m.m.f.  which  will  be  approximately  correct  for  any  given  power 
factor  of  the  external  load.  The  construction  is  shown  in  Fig. 
116,  and  since  vectors  are  used,  the  assumption  of  sine-wave 
functions  must  still  be  made.  This  is  where  an  error  is  intro- 
duced, because  the  distortion  of  the  flux  curves,  especially  with 


IR  Drop 

FIG.  115. — Vector  diagram  of  alternator  on  lagging  power  factor. 

salie.nt-pole  machines,  is  not  actually  in  accordance  with  this 
simple  law;  but  the  final  check  on  the  work  will  be  made  later 
when  the  flux  distribution  curves  are  plotted. 

The  vectors  01,  OEt  and  OE'g  have  the  same  meaning  as  in 
Fig.  115,  the  component  PE'g  being  the  total  reactive  voltage 
drop  both  of  end  connections  (formula  99)  and  slot  leakage 
(formula  106).  Draw  the  vector  OM  in  phase  with  OE'g  to 
represent  the  resultant  m.m.f.  necessary  to  overcome  air  gap 
and  tooth  reluctance  when  the  air-gap  flux  is  such  as  would 
develop  OE'g  volts  per  phase  in  the  armature  if  it  were  cut  by  all 
the  conductors.  If  we  neglect  the  effect  of  increased  tooth 
saturation,  this  m.m.f.  can  be  expressed  as 

riTpf 

OM  =  (open  circuit  SI  per  pole)  X  ~^ET 

\JEjt 

the  open-circuit  field  excitation  being  calculated  as  explained  in 
Arts.  92  and  93.  Now  draw  OMa  exactly  90  degrees  behind  01, 
to  represent  the  maximum  value  of  the  armature  m.m.f.  (formula 

19 


290          PRINCIPLES  OF  ELECTRICAL  DESIGN 

100).  This  must  be  balanced  by  the  field  component  MM0, 
giving  OM0  as  the  required  field  excitation  at  full  load.  If  the 
load  is  now  thrown  off,  the  developed  voltage  will  be  OE0,  where 
the  point  E0  is  the  intersection  of  OM0  and  the  prolongation  of 

the  line  PE'g,  because  this  satisfies  the  condition  ~°,   =  ~Qjnr' 

The  maximum  value  of  the  e.m.f.  OE0  will  be  generated  in  the 
conductors  immediately  opposite  the  center  of  the  pole  face. 
The  required  angle  of  displacement,  ft,  between  center  line  of 
pole  and  position  of  conductor  carrying  the  maximum  current 
may  thus  be  calculated,  and  the  full-load  flux  curves  plotted  as 
in  the  case  of  the  D.C.  machine,  where  the  displacement  of  the 
curve  of  armature  m.m.f.  is  determined  by  the  movement  of  the 
brushes.  It  must  not  be  overlooked  that  this  method  is  not 
strictly  accurate,  since  it  is  based  on  assumptions  that  are  rarely 
justified  in  practice. 

99.  Air-gap  Flux  Distribution  under  Load. — Having  deter- 
mined the  value  of  the  angle  ft  (Fig.  108),  the  curve  of  resultant 


FIG.  116. — Vector  diagram  of  alternator  m.m.fs. 

m.m.f.  for  any  condition  of  loading  can  be  obtained  by  adding 
the  ordinates  of  the  field  and  armature  m.m.f.  curves.  The 
procedure  is  then  the  same  as  was  followed  in  the  D.C.  design 
to  obtain  the  load  flux  curve  C  (Art.  43,  Chap.  VII),  except  that 
the  drawing  of  the  flux  curve  B  as  an  intermediate  step  will 
not  now  be  necessary,  seeing  that  the  effect  of  armature  distor- 
tion and  demagnetization  has  been  taken  account  of  in  the  vector 
construction  of  Fig.  116.  The  final  check  is  obtained  by  measur- 
ing the  area  of  flux  curve  C,  which  must  satisfy  the  condition 

Area  of  full-load  flux  curve  C         _  OE'g 
Area  of  open-circuit  flux  curve  A  ~  OEt 


AIR-GAP  FLUX  DISTRIBUTION  291 

If  the  approximate  value  of  the  field  ampere-turns,  as  given 
by  the  vector  OM0  of  Fig.  116,  does  not  produce  the  proper 
amount  of  flux  in  the  air  gap,  a  correction  must  be  made,  and  a 
new  curve  of  resultant  m.m.f.  obtained,  from  which  the  correct 
full-load  flux  curve  is  plotted. 

100.  Form  of  Developed  E.m.f.  Wave. — Having  plotted  the 
curve  of  air-gap  flux  distribution  for  any  given  condition  of 
loading,  it  is  an  easy  matter  to  obtain  a  curve  of  e.m.f.  due  to  the 
cutting  of  the  flux  by  the  armature  conductors.  It  may  be 
argued  that  it  is  not  quite  correct  to  derive  the  e.m.f.  wave  from 
the  curve  of  air-gap  flux  distribution,  because  the  flux  actually 
cut  by  each  armature  conductor  at  a  given  instant  depends  not 
only  upon  the  value  of  the  air-gap  density,  but  also  on  the  amount 
of  the  slot  leakage  flux  which  is  not  cut  by  the  conductor.  By 
referring  to  Fig.  Ill  (page  283)  it  will  be  seen  that,  although  the 
slot  leakage  appears  at  first  sight  to  pass  between  the  pole  and 
the  conductor,  it  actually  enters  the  armature  core  through  the 
teeth,  and,  with  the  exception  of  the  slot  flux  in  the  neutral  zone, 
it  all  links  with  the  armature  winding.  The  shape  of  the  e.m.f. 
wave  is  therefore  not  modified  to  any  great  extent  by  the  slot 
leakage  flux;  but,  unless  the  armature  current  is  zero  in  the  con- 
ductors passing  through  the  neutral  zone,  the  average  value  of 
the  developed  voltage  must  be  less  than  it  would  be  if  all  the 
flux  entering  the  tops  of  the  teeth  were  cut  by  the  conductors. 
This  is  shown  in  the  diagram,  Fig.  115,  where  QE'g  is  the  "  appar- 
ent" developed  voltage  (assuming  all  the  flux  lines  in  the  air 
gap  to  be  cut),  and  OEg  is  the  actual  developed  voltage.  It  is 
unnecessary  to  introduce  refinements  with  a  view  to  determining 
the  exact  wave  shape  of  the  e.m.f.  actually  developed  in  the  con- 
ductors because,  by  using  the  flux  curve  C  of  air-gap  distribution, 
the  wave  shape  of  the  ''apparent"  developed  e.m.f.  is  obtained, 
and  with  the  aid  of  equivalent  sine-waves  (to  be  explained  later) 
the  terminal  voltage  can  be  calculated  with  sufficient  accuracy 
for  practical  purposes.  It  is  important  to  bear  in  mind  that  the 
e.m.f.  wave-shape  obtained  at  the  terminals  of  a  Y-connected 
three-phrase  generator  is  not  necessarily  the  same  as  the  wave 
shape  developed  in  each  phase-winding  by  the  cutting  of  the 
flux  in  the  air  gap.  This  was  explained  in  Art.  71  (page  246), 
and  in  order  to  obtain  the  wave-form  of  e.m.f.  at  the  terminals 
of  a  Y-connected  generator,  it  is  necessary  to  add  the  corres- 
ponding ordinates  of  two  star-voltage  waves  plotted  with  a 


292 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


phase  displacement  of  one-third  of  a  pole  pitch  (i.e.  60  electrical 
degrees). 

Whatever  may  be  the  number  and  spacing  of  the  slots  on  the 
armature  surface,  the  usual  e.m.f.  wave  shapes  can  always  be 
plotted  in  the  manner  indicated  in  Fig.  117  where  the  full  line 
curve  may  be  any  one  of  the  flux  curves  previously  obtained, 
the  ordinates  of  which  are  a  measure  of  the  flux  density  in  the 
air  space  near  the  surface  of  the  armature.  Draw  A  and  B 
properly  spaced  to  represent  the  slots,  e.g.,  two  in  this  illustra- 
tion, of  one  phase  of  the  armature  winding.  At  the  instant  when 
the  center  line  C  of  this  phase  winding  occupies  the  position 
shown  in  Fig.  117,  the  conductors  in  slot  A  are  moving  in  a 
field  of  density  A  A',  while  the  conductors  in  slot  B  are  moving 


rve 


FIG.  117.  —  Method  of  deriving  e.m.f.  wave  from  curve  of  air-gap  flux 

distribution. 

in  a  field  of  density  BBf.     These  conductors  are  all  in  series,  and 
the  instantaneous  value  of  the  voltage  per  phase  winding  will  be 


60  X  10* 

where  Ba  —  average  value  of  flux  density,  in  gausses; 

AA'  +  BB'  .     .  '     . 
=  -     —  ^—     -  in  this  instance. 

2i 

N  =  revolutions  per  minute. 
D  =  armature  diameter  in  centimeters. 
la  =  armature  length  (axial)  in  centimeters. 
Z  =  total  number  of  conductors  in  series  per  phase. 

This  value  of  e  is  plotted  as  CC'  to  a  suitable  scale,  and  the  process 
is  repeated  for  other  positions  of  the  armature  slots,  thus  pro- 
ducing the  dotted  curve  7,  representing  the  electromotive  force 


AIR-GAP  FLUX  DISTRIBUTION 


293 


that  would  be  developed  in  the  windings  if  all  the  flux  in  the  air 
gap  were  cut  by  the  conductors  in  the  slots. 

The  general  solution,  which  includes  fractional  pitch  windings, 
is  illustrated  in  Fig.  118.  The  instantaneous  value  of  the  aver- 
age flux  density  for  n  slots  per  pole  per  phase  is 


Bn     = 


(a  +  b  +  c  + 


)  -  (a'  +  V  +  c'+ 


2n 


The  relative  positions  of  the  slots  and  the  center  of  the  coil  (P) 
may  be  marked  on  a  separate  strip  of  paper  that  can  be  moved 
to  any  desired  position  under  the  flux  curve;  and  the  instan- 


Curve  of  Volts  (  6  )  Plotted 

Relatively  to  Position  of 

Center  of  Armature 

A 


\ 


\ 


\ 


\ 


FIG.  118. — Flux  curve  and  resulting  e.m.f.  wave — fractional  pitch 
armature  winding. 

taneous  values  of  the  voltage  can  then  conveniently  be  plotted 
over  the  point  P.     For  this  instantaneous  voltage  we  may  write 

Average  instantaneous 
e.m.f.  per  conductor 

=  flux  cut  per  centimeter  of  travel 
X  centimeters  per  second  X  10~8 

=  (Bah)  X  v  X  10~8 


where  v 


60 


cm.   per  second.     The  instantaneous  voltage 


per  phase  is  therefore 

e  =  ec  X  z  = 
as  stated  in  formula  (107). 


60  X108 


294 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


This  step-by-step  method  of  drawing  the  e.m.f .  waves  will  yield 
surprisingly  accurate  results,  with  the  one  exception  that  the 
ripples  known  as  "tooth  harmonics"  which  are  generally  present 
in  oscillograph  records,  will  not  appear  in  the  graphical  work. 
The  effect  of  the  distributed  winding  in  smoothing  out  the  irregu- 
larities of  the  flux-distribution  curve  is  very  clearly  shown  by  the 
shape  of  th.e  e.m.f.  wave  in  Fig.  118. 

101.  Form  Factor.— The  ratio  of  the  r.m.s.  or  virtual  value  to 
the  mean  value  of  an  alternating  e.m.f.  or  current  is  the  form 
factor.  The  average  ordinate  of  an  irregular  wave  such  as  may 
be  obtained  by  the  process  represented  in  Figs.  117  and  118,  is 
readily  obtained  by  measuring  its  area  with  a  planimeter  and 


FIG.  119. — Illustrating  calculation  FIG.  120. — Wave  of  alternating 
of  r.m.s.  value  of  variable  quantity  e.m.f.  plotted  to  polar  coordinates, 
plotted  to  polar  coordinates. 

then  dividing  this  area  by  the  length  of  the  base  line,  i.e.,  the  pole 
pitch.  If  another  curve  is  plotted  by  squaring  the  ordinates  of 
the  original  curve,  it  is  merely  necessary  to  take  the  square  root 
of  the  average  ordinate  of  this  new  curve  in  order  to  obtain  the 
virtual  value  of  the  alternating  quantity.  It  will,  however, 
be  more  convenient  to  re-plot  the  original  curve  to  polar  co- 
ordinates. The  general  case  of  a  variable  quantity  plotted  to 
polar  coordinates  is  illustrated  in  Fig.  119,  where  the  radial 
distance  from  the  point  0  represents  the  instantaneous  value  of 
the  variable  quantity,  while  time  (or  distance  of  travel)  is 
measured  by  the  angular  distance  between  the  vector  considered 
and  the  axis  OX. 


AIR-GAP  FLUX  DISTRIBUTION  295 

If  r  be  the  length  of  the  vector,  and  8  the  angular  distance  from 
the  reference  axis,  we  may  write 

Area  of  triangle  OSP  =  Y^r  X  rdB 


and  the  area  included  between  any  given  angular  limits  ft  and 
a  is 

Area  OAB  (shaded)  ==  2^  %r*dO 

=  %  (average  value  of  r2)  X  03  —  «) 
twice  area  of  curve 


whence  average  value  of  r2 


Applying  this  rule  to  the  case  of  a  periodically  varying  e.m.f. 
or  current,  we  have  in  Fig.  120  a  representation  of  an  e.m.f. 
wave  plotted  to  polar  coordnates.  This  may  be  thought  of  as  the 
actual  e.m.f.  wave  obtained  by  the  graphical  method  previously 
outlined,  but  transferred  from  rectangular  to  polar  coordinates. 
The  radius  vector  (moving  in  a  counter-clockwise  direction, 
covers  the  complete  area  of  one  lobe  when  it  has  moved  through 
an  angle  of  180  degrees;  because,  in  this  diagram,  the  electrical 
degrees  are  correctly  represented  by  the  actual  space  degrees.' 
The  angle  moved  through  during  the  half  period  is  TT  radians,  and 
the  virtual  value  of  the  alternating  e.m.f.  is  therefore 


T-J          /2  (area  of  one  lobe) 

E '-'-   ~    - 


The  area  of  the  curve  is  easily  measured  with  a  planimeter, 
and  the  value  of  E  thus  obtained  has  merely  to  be  divided  by  the 
previously  calculated  average  value  in  order  to  obtain  the  form 
factor  of  the  irregular  wave. 

102.  Equivalent  Sine -waves. — Equivalent  sine-waves  are  a 
great  convenience  in  power  calculations  because  they  permit  the 
use  of  vectors,  and  enable  us  to  express  the  power  factor  as  the 
cosine  of  a  definite  angle.  Whenever  vector  diagrams  are  used, 
the  alternating  quantities  must  be  sine  functions  of  time;  and 
when  applied  to  practical  calculations  involving  irregular  (i.e., 
non-sine)  wave  shapes,  they  must  be  thought  of  as  representing 
"equivalent"  sine  functions.  It  will,  of  cour  e  be  understood 
that,  in  many  cases,  the  substitution  of  a  sine  curve  for  the  actual 
wave  form  is  not  permissible;  the  effects,  for  instance,  of  the 
higher  harmonics  on  a  condensive  load  cannot  be  annulled  by 


296 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


imagining  the  actual  wave  to  be  replaced  by  a  so-called  equiva- 
lent smooth  wave;  but  the  use  of  equivalent  sine- waves  for 
power  calculations  on  practical  A.C.  circuits,  can  generally  be 
justified.  An  equivalent  sine- wave  may  be  defined  as  a  sine- 
wave  of  the  same  periodicity  and  the  same  virtual  value  as  the 
irregular  wave  which  it  is  supposed  to  replace.  An  equivalent 
sine-wave  of  current  would  produce  the  same  heating  effects  as 
the  irregular  wave;  but  its  mean  value,  and  therefore  its  form 
factor,  may  be  different. 

A  sine-wave  plotted  to  polar  coordinates  will  be  a  circle,  of 
which  the  diameter — representing  the  maximum  value  of  the 
sine-wave — is  easily  calculated  since  the  equivalent  wave  must 


FIG.  121. — Irregular  wave  and  equivalent  sine  wave  plotted  to  polar 

coordinates. 

have  the  same  root-mean-square  value  as  the  non-sinusoidal 
wave,  and  therefore  also  the  same  area  when  plotted  to  polar 
coordinates. 

Let  d  =  the  diameter  of  the  equivalent  circle  (or  maximum 

value  of  the  equivalent  sine -wave) 
and  let  A  =  the  area  of  one  lobe  of  the  irregular  wave  plotted  to 

polar  coordinates, 
then 


whence 


d  = 


(108) 


AIR-GAP  FLUX  DISTRIBUTION 


297 


The  next  point  to  consider  is  the  position  of  the  equivalent 
sine-wave  of  maximum  ordinate  d,  relatively  to  some  particular 
value  of  the  irregular  wave.  It  is  obvious  that  neither  the 
maximum  nor  the  zero  value  of  the  two  waves  must  necessarily 
coincide;  but  by  so  placing  the  equivalent  sine-wave  relatively  to 
the  irregular  wave  that  each  quarter  wave  of  the  one  has  the 
same  virtual  value  as  the  corresponding  quarter  wave  of  the 
other,  the  proper  position  of  the  equivalent  wave  may  be 
determined.  This  will  be  better  understood  by  referring  to 
Fig.  121. 

The  irregular  wave  is  plotted  to  polar  coordinates,  and  its 
area  measured  with  the  aid  of  a  planimeter.  The  line  OM  is  then 
drawn,  dividing  this  area  in  two  equal  parts.  This  is  easily 
done  with  the  help  of  the  planimeter,  the  proper  position  of  the 
dividing  line  being  found  when  the  shaded  area  of  the  irregular 


Actual  E.M.F.  Wave 


Equivalent  Sine 
Wave 


Geometric » 

Neutral   f~ 


FIG.  122. — Curves  of  Fig.  121  re-plotted  to  rectangular  coordinates. 

wave  is  exactly  equal  to  the  unshaded  area.  It  is  upon  this  line 
(OM)  that  the  center  of  the  equivalent  circle  of  diameter  d  (see 
formula  108)  must  be  placed.  If  the  irregular  wave  has  been 
correctly  plotted  relatively  to  some  reference  axis,  such  as  the 
geometrical  neutral  line,  or  the  pole  center,  the  angle  a  can  be 
measured.  This  angle  represents  the  displacement  of  the 
maximum  value  of  the  equivalent  wave  beyond  the  pole  center, 
and  when  used  in  connection  with  the  irregular  wave,  it  may  be 
thought  of  as  the  average  displacement  of  the  distorted  e.m.f. 
behind  the  position  of  open-circuit  e.m.f.,  which  will  be  sym- 
metrically placed  about  the  center  line  of  the  pole.  This  angle  a 
has  the  same  meaning  as  the  angle  M0OM  of  Fig.  116;  but  it  has 
now  been  determined  with  greater  accuracy  than  could  be 


298          PRINCIPLES  OF  ELECTRICAL  DESIGN 

expected  of  the  vector  construction,  in  which  the  loss  of  pressure 
due  to  armature  distortion  was  assumed  to  be  in  accordance  with 
the  sine  law.  Fig.  122  illustrates  the  same  condition  as  Fig. 
121  except  that  the  e.m.f.  waves  have  been  re-plotted  to  rec- 
tangular coordinates.  The  practical  application  of  equivalent 
sine -waves  in  predetermining  the  regulation  of  an  alternating- 
current  generator  will  be  taken  up  in  the  following  chapter,  and 
again  in  Chap.  XV,  when  working  out  a  numerical  example. 


CHAPTER  XIV 
REGULATION  AND  EFFICIENCY  OF  ALTERNATORS 

103.  The  Magnetic  Circuit. — Except  for  the  fact  that  the  field 
magnets  usually  rotate,  the  design  of  the  complete  magnetic  cir- 
cuit of  an  alternating-current  generator  differs  little  from  that  of 
a  D.C.  dynamo.  Given  the  ampere-turns  required  per  pole,  and 
the  voltage  of  the  continuous-current  circuit  from  which  the  ex- 
citing current  is  obtained  (usually  about  125  volts),  the  procedure 
for  calculating  the  size  of  wire  required  is  the  same  as  would  be 
followed  in  designing  any  other  shunt  coils  (see  Art.  10,  Chap.  II 
and  Art.  58,  Chap  IX).  When  estimating  the  voltage  per  pole 
across  the  field  winding,  a  suitable  allowance  must  be  made  for 
the  pressure  absorbed  by  the  rheostat  in  series  with  the  field 
windings.  The  exact  amount  of  excitation  required  under  any 
given  condition  of  loading  can,  of  course,  be  determined  only 
after  the  complete  magnetic  circuit  has  been  designed. 

A  higher  current  density  may  be  allowed  in  the  copper  of  rotat- 
ing field  coils  than  in  stationary  coils  against  which  air  is  thrown 
by  the  rotation  of  the  armature,  because  the  cooling  is  more  ef- 
fective, the  difference  being  especially  noticeable  at  the  higher 
peripheral  speeds.  In  the  absence  of  reliable  data  on  any  par- 
ticular type  and  size  of  machine,  the  curve  of  Fig.  123  may  be 
used  for  selecting  a  suitable  cooling  coefficient.  The  cooling  sur- 
face considered  includes,  as  before,  the  inside  surface  near  the 
pole  core  and  the  two  ends,  in  addition  to  the  outside  surface  of 
the  coil.  It  is  to  be  understood  that  the  cooling  coefficient  ob- 
tained from  Fig.  123  is  approximate  only,  being  an  average  of 
many  tests  on  different  sizes  and  shapes  of  coils  on  rotating  field 
magnets. 

In  determining  the  amount  by  which  the  pole  must  project 
from  the  yoke  ring,  it  is  well  to  allow  about  1  in.  of  radial  length 
of  winding  space  for  every  1,500  ampere-turns  per  pole  required 
at  full  load  (i.e.,  estimated  maximum  excitation).  An  effort 
should  be  made  to  keep  the  radial  projection  of  the  poles  as  small 
as  possible  in  order  to  prevent  excessive  magnetic  leakage.  A 

299 


300 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


radial  length  of  pole  greater  than  two  or  two  and  one-half 
times  the  width  of  pole  core  (measured  circumferentially)  would 
be  a  poor  design, 'because  the  gain  in  winding  space  due  to  increase 
of  radial  length  would  be  largely  neutralized  by  the  greater  amount 
of  flux  per  pole  due  to  leakage. 

The  required  useful  flux  per  pole  being  known,  the  flux  to  be 
carried  by  pole  core  and  yoke  may  be  calculated  if  the  leakage 
factor  is  known.  The  calculation  of  the  permeance  of  the  air 


U.UZS 

0.022 
g  0.021 

3  0.020 
g 
8  0.019 

8 

|  0.018 

Q 
$3  0.017 

•S  0.016 

f  °'015 

P. 

§  0.014 
^  0.013 

4J 

|  0.012 
J  0.011 
J3  0.010 
^  0.009 
0.008 

/ 

/ 

/ 

/ 

/ 

/ 

> 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

1000          2000  3000  4000  5000          6000  7000          8000 

Peripheral  Speed  of  (Salient  Pole)  Rotor-  Feet  per  Minute 

FIG.  123. — Cooling  coefficient  for  field  windings  of  rotating-field  alternators. 

paths  between  poles  is  tedious  and  somewhat  unsatisfactory.  It 
seems  therefore  best,  in  designs  of  normal  types,  to  assume  a 
leakage  factor  based  on  measurements  made  on  existing  machines. 
The  leakage  coefficient  will  be  low  in  machines  with  large  pole 
pitch,  and  high  in  the  case  of  slow-speed  engine-driven  genera- 
tors with  a  large  number  of  closely  spaced  poles.  The  following 
approximate  values  may  be  used  for  estimating  the  flux  in  poles 
and  yoke  ring. 


REGULATION  OF  ALTERNATORS  301 


High  values;  to  be  selected  when  pole  pitch  is  small, 
and  radial  length  of  pole  core  great  in  proportion  to 


1.32  to  1.42 


1.22  to  1.32 


width. 
Average  values;  for  pole  pitch  8  to  12  in.,  and  length 

of  winding  space  about  equal  to  width  of  pole  core. 
Low  values;  for  large  pole  pitch  and  small  radial  \ 

i  .1        e         i  (    1  •  1O  tO  1  .  A* 

length  of  pole  core. 

These  leakage  coefficients  apply  to  the  case  of  alternators  with 
field  excitation  to  give  approximately  normal  voltage  at  terminals 
on  open  circuit. 

Provided  a  reasonably  high  leakage  factor  has  been  used,  the 
cross-section  of  the  poles  and  yoke  of  good  dynamo  steel  may  be 
calculated  for  a  flux  density  up  to  15,500  gausses.  Although 
the  flux  density  in  the  pole  core  (of  uniform  cross-section)  will 
fall  off  in  value  as  the  distance  from  the  yoke  ring  increases,  the 
effect  of  the  distributed  leakage  may  be  taken  care  of  by  calcu- 
lating the  ampere-turns  for  the  pole  core  on  the  assumption  that 
the  total  leakage  flux  is  carried  by  the  pole  core,  but  that  the 
length  of  the  pole  is  reduced  to  half  its  actual  value. 

Bearing  in  mind  the  above-mentioned  points,  the  open-circuit 
saturation  curve — connecting  ampere-turns  per  pole  and  resulting 
terminal  voltage — can  be  calculated  and  plotted  exactly  as  in  the 
case  of  a  continuous-current  dynamo  with  rotating  armature  and 
stationary  poles  (see  Art.  57,  Chap.  IX,  and  item  128  of  Art.  63, 
Chap.  X). 

104.  Regulation. — Reference  has  already  been  made  in  Art.  77 
of  Chap.  XI  to  the  regulation  of  alternating-current  generators, 
and  it  was  pointed  out  that  the  designer  does  not  always  aim  at 
producing  a  machine  with  a  high  percentage  inherent  regulation, 
because  it  is  recognized  that  automatic  field  regulation  or  some 
equivalent  means  of  varying  the  field  ampere-turns  is  necessary 
in  order  to  maintain  the  proper  terminal  voltage  under  varying 
conditions  of  load  and  power  factor.  The  large  modern  units 
driven  at  high  speeds  by  steam  turbines  are,  indeed,  purposely 
designed  to  have  large  armature  reactance  in  order  to  limit  the 
short-circuit  current,  the  maximum  value  of  which — at  the 
instant  the  short-circuit  occurs — depends  rather  upon  the  arma- 
ture reactance  than  upon  the  demagnetizing  or  distortional  effect 
of  the  armature  current.  These  considerations  tend  to  empha- 
size the  importance  of  correctly  predetermining  the  inherent 
regulation  of  machines,  and  it  is  important  to  know  exactly 
what  the  term  " armature  reactance"  should  include,  in  order 


302 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


that  the  probable  short-circuit  current  may  be  estimated,  not 
only  after  the  armature  ampere-turns  have  had  time  to  react 
upon  the  exciting  field,  but  also  at  the  instant  when  the  impedance 
of  the  armature  windings  alone  limits  the  current. 

The  usual  methods  of  predetermining  the  regulation  of  alter- 
nators involve  almost  invariably  the  use  of  vectors  or  vector 
algebra.  This  is  convenient,  and  to  some  extent  helpful,  be- 
cause the  problem  is  thus  presented  in  its  simplest  aspect:  the 
very  fact  that  vectors  are  used  assumes  the  sinusoidal  variation 
of  the  alternating  quantities,  or  the  substitution  of  so-called 

equiyalent  sine-wave  forms  for 
the  actual  wave  shapes,  thus 
eliminating  the  less  easily  cal- 
culated effects  caused  by  cross- 
magnetization  and  the  conse- 
quent distortion  of  the  wave 
shapes.  On  the  other  hand,  the 
omission  of  these  factors,  espe- 
cially when  departures  are  made 
from  standard  types,  may  lead 
to  incorrect  conclusions,  and  in 
any  case  the  plotting  of  the  ac- 
tual flux  and  e.m.f.  curves  is 
of  great  value  to  the  designer. 

Ampere- turns  on  field    H       »       .     It  is  therefore  proposed  to  con- 
FIQ.  124. — Inherent  regulation  ob-  sider  in  the  first  place  what  is 

the  best  that  Can  be  done  with 
the  aid  of  vectors  on  the  usual 

assumption  of  sine-wave  form,  and  afterward  show  how  a  greater 
degree  of  accuracy  can  be  attained  by  using  curves  representing 
the  actual  flux  distribution  in  the  air  gap,  corresponding  to  the 
required  load  conditions. 

The  curves  in  Fig.  124  may  be  considered  as  having  been 
plotted  from  actual  test  data.  The  upper  curve  is  the  open- 
circuit  saturation  characteristic,  giving  the  relation  between  the 
number  of  ampere-turns  of  field  excitation  per  pole  and  the  pres- 
sure at  the  terminals,  which  in  this  case  is  the  same  as  the  electro- 
motive force  actually  developed  in  the  armature  windings.  The 
lower  curve  is  the  load  characteristic  corresponding  to  a  given 
armature  current  and  a  given  external  power  factor.  The  in- 
herent regulation  when  the  field  excitation  is  of  the  constant 


REGULATION  OF  ALTERNATORS  303 

value  OP  is  therefore  the  difference  of  terminal  voltage  E0Ei 
divided  by  the  load  voltage  EI,  or,  expressed  as  a  percentage  of 
the  lower  voltage, 

OE0  -  OEt 
OEi 

Thus  the  error  in  predetermining  the  inherent  regulation  de- 
pends upon  the  degree  of  accuracy  within  which  curves  such  as 
those  shown  in  Fig.  124  can  be  drawn  before  the  machine  has 
been  built  and  tested. 

105.  Factors  Influencing  the  Inherent  Regulation  of  Alter- 
nators.— By  enumerating  all  the  factors  which  influence  the 
terminal  voltage  of  a  generator  driven  at  constant  speed  with 
constant  field  excitation,  it  will  be  possible  to  judge  how  nearly 
the  methods  about  to  be  considered  approximate  to  the  ideal 
solution  of  the  problem.  These  factors  are: 

(a)  The  total  or  resultant  flux  actually  cut  by  the  armature 
windings  (this  involves  the  flux  linkages  producing  armature 
reactance). 

(6)  The  ohmic  resistance  of  the  armature  windings. 

(c)  The  alteration  in  wave  shape  of  the  generated  electro- 
motive force,  due  to  changes  in  air-gap  flux  distribution.  This 
means  that  the  measured  terminal  voltage  depends  not  only 
upon  the  amount  of  flux  cut  by  the  conductors  but  also  upon  the 
distribution  of  flux  over  the  pole  pitch,  because  the  amount  of 
flux  cut.determines  the  average  value  of  the  developed  voltage, 
while  the  form  of  the  e.m.f .  wave  determines  the  relation  between 
the  mean  value  and  the  virtual  or  r.m.s.  value. 

By  far  the  most  important  items  are  included  under  (a),  and 
it  will  be  well  to  consider  exactly  how  the  resultant  flux  cut  by 
the  armature  windings  varies  when  load  is  put  on  the  machine. 

Considering  first  the  flux  cut  by  the  active  belt  of  conductors 
under  the  pole  face,  this  is  not  usually  the  same  under  load  con- 
ditions as  on  open  circuit  (the  field  excitation  remaining  con- 
stant), for  the  following  reasons.  The  current  in  the  armature 
windings  produces  a  magnetizing  effect  which,  together  with  the 
field-pole  magnetomotive  force,  determines  the  resultant  mag- 
netomotive force  and  the  actual  distribution  of  the  flux  in  the  air 
gap.  When  the  power  factor  of  the  load  is  approximately  unity, 
the  armature  current  produces  cross-magnetization  and  dis- 
tortion of  the  resultant  field,  accompanied  usually  by  a  re- 
duction of  the  total  flux  owing  to  increased  flux  density  in  the 


304          PRINCIPLES  OF  ELECTRICAL  DESIGN 

armature  teeth  where  the  air-gap  density  is  greatest.  The  effect 
is,  however,  less  marked  in  alternating-current  than  in  con- 
tinuous-current generators,  because  in  the  former  the  tooth 
density  is  rarely  so  high  as  to  approach  saturation.  On  low 
power  factor,  with  lagging  current,  the  armature  magnetomotive 
force  tends  to  oppose  the  field  magnetomotive  force,  and  on  zero 
power  factor  its  effect  is  wholly  demagnetizing,  thus  greatly  re- 
ducing the  resultant  air-gap  flux.  With  a  leading  current  the 
well-known  effect  of  an  increased  flux  and  a  higher  voltage  is 
obtained.  The  effect  known  as  armature  reaction,  as  distin- 
guished from  armature  reactance,  is  therefore  dependent  not 
only  on  the  amount  of  the  armature  current  but  also  largely 
upon  the  power  factor. 

The  effect  of  the  individual  conductors  in  producing  slot 
leakage  was  discussed  in  Art.  95  of  Chap.  XIII,  and  illustrated 
by  Figs.  110  and  111,  wherein  it  is  clearly  shown  that,  as  current 
is  taken  out  of  the  armature,  the  total  flux  cut  by  the  active 
conductors  is  less  than  at  no  load  (with  the  same  field  excitation) 
by  the  amount  of  the  slot  flux — or  equivalent  slot  flux — which 
passes  from  tooth  to  tooth  in  the  neutral  zone. 

Turning  now  to  the  flux  cut  by  the  end  connections,  i.e.,  by 
those  portions  of  the  armature  winding  which  project  beyond 
the  ends  of  the  slots,  this  flux  is  set  up  almost  entirely  by  the 
magnetomotive  force  of  the  armature  windings,  and  is  negligible 
on  open  circuit.  For  a  given  output  and  power  factor,  the  end 
flux  in  a  polyphase  generator  is  fixed  in  position  relatively  to  the 
field  poles,  being  stationary  in  space  if  the  armature  revolves. 
The  maximum  value  of  the  armature  magnetomotive  force  occurs 
at  the  point  where  the  current  in  the  conductors  is  zero,  and  on 
the  assumption  of  a  sinusoidal  flux  distribution,  the  electromotive 
force  generated  by  the  cutting  of  these  end  fluxes  may  be  repre- 
sented correctly  as  a  vector  drawn  90  degrees  behind  the  current 
vector.  It  is  therefore  permissible  to  consider  this  e.m.f.  com- 
ponent as  a  reactive  voltage  such  as  would  be  obtained  by  con- 
necting a  choking  coil  in  series  with  the  " active"  portion  of  the 
armature  windings;  and  if  the  inductance,  Le,  of  the  end  windings 
is  known,  and  a  sinusoidal  flux  distribution  assumed,  the  electro- 
motive force  developed  by  the  cutting  of  the  end  fluxes  under 
load  conditions  is  given  by  the  well-known  expression  2irfLeIc, 
where  Ic  is  the  virtual  value  of  the  current  in  the  armature  wind- 
ings, and  /  is  the  frequency. 


REGULATION  OF  ALTERNATORS  305 

This  quantity  was  calculated  in  Art.  87,  Chap.  XII,  and  ex- 
pressed in  formula  (99),  the  calculation  being  based  upon  an 
amount  of  end  flux  per  pole  ($f)  given  by  the  empirical  formula 
(98).  Although  the  writer  likes  to  think  of  the  cutting  of  the 
end  flux  by  the  conductors  projecting  beyond  the  ends  of  the 
slots,  the  idea  of  flux-linkages  and  a  coefficient  of  self-induction, 
Lf,  expressed  in  henrys,  may  be  preferred  by  others.  If  it  is  de- 
sired to  substitute  the  terms  of  the  formulas  (98)  and  (99)  in  the 
expression  2irfLeIc,  the  value  of  the  coefficient  of  self-induction, 
in  henrys,  will  be 


X  10* 


106.  Regulation  on  Zero  Power  Factor.  —  In  practice,  any 
power  factor  below  20  per  cent,  is  usually  considered  to  be  equiva- 
lent to  zero,  so  that  the  calculations  can  be  checked  when  the 
machine  is  built,  by  providing  as  a  load  for  the  generator  a  suit- 
able number  of  induction  motors  running  light.  On  these  low 
power  factors  with  lagging  current  the  phase  displacement  of  the 
armature  current  causes  the  armature  magnetomotive  force  to 
be  almost  wholly  demagnetizing,  that  is  to  say,  it  directly  opposes 
the  magnetomotive  force  due  to  the  field  windings,  the  distor- 
tional  or  cross-magnetizing  effect  being  negligible.  Its  maximum 
value  per  pole  is  given  by  formulas  (100)  and  101)  of  Art.  94, 
Chap.  XIII,  and  its  effect  in  reducing  the  flux  in  the  air  gap  is 
readily  compensated  (on  zero  power  factor)  by  increasing  the 
field  excitation  so  that  the  resultant  ampere-turns  remain  un- 
changed. This  statement  is  not  strictly  correct  because  the  in- 
creased ampere-turns  on  the  field  poles  give  rise  to  a  greater  leak- 
age flux,  and  this  alteration  should  not  be  overlooked,  especially 
when  working  with  high  flux  densities  in  the  iron  of  the  magnetic 
circuit.  If  the  estimated  leakage  flux  for  a  given  developed 
voltage  on  open  circuit  is  fy  maxwells,  then,  for  the  same  voltage 
with  full-load  current  on  zero  power  factor,  the  leakage  flux  would 

be  approximately  &i  =  <£*  (~~~/fr  —  ;  where  M  is  the  number  of 

field  ampere-turns  on  open  circuit,  and  (M  +  Ma)  is  the  number 
of  field  ampere-turns  with  full-load  current  in  the  armature,  the 
power  factor  being  zero.  The  quantity  Ma  is  the  demagnetizing 
ampere-turns  per  pole  due  to  the  armature  current. 

Let  curve  A  of  Fig.  125  be  the  open-circuit  saturation  curve 
20 


306 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


of  the  machine,  referred  to  in  Art.  103 ;  it  is  the  curve  for  the  com- 
plete machine,  and  can  be  plotted  only  after  the  magnetic  circuit 
external  to  the  armature  has  been  designed. 

Knowing  the  increase  of  flux  in  pole  and  frame  the  magneto- 
motive force  absorbed  in  overcoming  the  increased  reluctance  of 
these  parts  can  be  calculated,  and  in  this  way  the  dotted  curve 
A'  of  Fig.  125  can  be  drawn.  This  is  merely  the  open-circuit 
saturation  curve  corrected  for  increased  leakage  flux  due  to  the 
additional  field  current  required  to  balance  the  demagnetizing 
effect  of  a  given  armature  current. 

Assuming  all  the  alternating  quantities  to  be  simple  harmonic 

functions,  the  vector  diagram 
(Fig.  126)  can  be  drawn  as  ex- 
plained in  Art.  96.  It  shows 
the  voltage  components  for  one 
.phase  of  the  winding;  the  vector 
PEt  being  the  IR  drop,  while 
E0P  and  E'0Eg  stand  for  the  re- 
actance voltage  drops  due  to 
end  flux  and  slot  flux  respec- 
tively. The  numerical  value  of 
EgP  can  be  obtained  from  for- 
mula (99)  page  266,  and  oiE'0Eg 
from  formula  (106)  of  page  288. 
The  apparent  developed  volt- 

FIG.    125.— Methods  of  construct-  age  E'g  obtained  from  Fig.  126 
ing  suturation  curve  for  zero  power     .          ,,  .    ,     ,,         ,, 

factor.  gives  the  point  M  on  the  cor- 

rected no-load  saturation  curve 

A'  in  Fig.  125,  and  the  distance  E'gM  or  OM'  shows  what  exciting 
ampere-turns  are  required  to  develop  this  electromotive  force. 
The  terminal  voltage  is,  however,  only  Et,  which  gives  the  point 
N  of  the  triangle  MNR.  Now  draw  NR  parallel  to  the  horizontal 
axis  to  represent  the  total  number  of  ampere-turns  per  pole  due 
to  the^armature  current,  which,  as  previously  explained,  will  be 
entirely  demagnetizing,  and  must  therefore  be  compensated  by 
an  equal  number  of  ampere-turns  on  the  field  pole.  Thus  EtR 
or  OR'  is  the  field  excitation  necessary  to  produce  Et  volts  at  the 
terminals  of  the  machine.  If  the  load  is  now  thrown  off,  the 
terminal  pressure  will  rise  to  E0  and  the  percentage  regulation 
for  this  particular  current  output  on  zero  power  factor  will  there- 

SR 
fore  be  100    p  p,  •     This  simple  construction   enables  the  de- 


MR 

Ampere-turns   per    pole 


REGULATION  OF  ALTERNATORS  307 

signer  to  predetermine  with  but  little  error  the  regulation  on  zero 
power  factor  provided  he  can  correctly  calculate  the  reactances 
required  for  the  vector  quantities  of  Fig.  J.26.  The  complete 
load  characteristic  O'R  is  quickly  obtained  by  sliding  the  triangle 
MNR  along  the  corrected  no-load  saturation  curve.1  The  dif- 
ference of  pressure,  SR,  corresponding  to  any  particular  value 
OR'  of  field  excitation  (Fig.  125)  is  called  the  synchronous  react- 
ance drop  because,  although  it  is  made  up  partly  of  real  reactance 
drop  and  partly  of  armature  reaction,  it  may  conveniently  be 
treated  as  if  it  were  due  to  an  equivalent  or  fictitious  reactance 
capable  of  producing  the  same  total  loss  of  pressure  if  the  magneto- 
motive force  of  the  armature  had  no  demagnetizing  or  distortional 
effect.  Thus,  by  producing  the  line  PE'g  to  E0  in  Fig.  126,  so 
that  PE0  is  equal  to  RS  of  Fig.  125,  the  vector  diagram  shows  the 
difference  between  the  open-circuit  pressure  OE0  and  the  terminal 


(slots) 

• 

I 

FIG.  126. — Vector  diagram  for  zero  power  factor. 

pressure  OEt  under  load  conditions  at  zero  power  factor  when  the 
field  excitation  is  maintained  constant.  The  additional  (ficti- 
tious) reactance  drop  E0E'g  is  correctly  drawn  at  right  angles  to 
the  current  vector  because  on  zero  power  factor  the  effect  of  the 
armature  magnetomotive  force  is  wholly  demagnetizing;  in  other 
words,  it  tends  to  set  up  a  magnetic  field  displaced  exactly  90 
degrees  (electrical  space)  behind  the  current  producing  it;  hence 
when  the  load  is  thrown  off,  the  balancing  m.m.f.  component  on 
the  field  poles  will  generate  the  additional  voltage  in  the  phase 
OE0.  It  should  be  realized  that  the  fictitious  reactance  drop, 
E0E'g  of  Fig.  126,  cannot  be  predetermined  until  the  whole  of  the 

1  S.  H.  MORTENSEN.  "Regulation  of  Definite  Pole  Alternators."  Trans. 
A.  I.  E.  E.,  vol.  32,  p.  789, 1913.  Also  B.  A.  BEHREND.  "  The  Experimental 
Basis  for  the  Theory  of  the  Regulation  of  Alternators."  Trans.  A.  I.  E.  E., 
vol.  21,  p.  497,  1903;  and  B.  T.  McCoRMicx  in  discussion  on  "A  Con- 
tribution to  the  Theory  of  the  Regulation  of  Alternators"  (H.  M.  HOBART 
and  F.  PUNGA).  Ibid.,  vol.  23,  p.  330,  1904. 


308 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


ix(ends) 


IRdrop 


magnetic  circuit  of  the  machine  has  been  designed.  The  distance 
DR  in  Fig.  125  is  the  loss  of  voltage  corresponding  to  E'gP  of 
Fig.  126,  and  is  approximately  constant  for  a -given  armature 
current.  The  portion  SD,  however,  of  the  total  difference  of 
voltage  depends  on  the  slope  of  the  line  MS,  and  is  thus  some 
function  of  the  degree  of  saturation  of  the  iron  in  the  magnetic 
circuit.  It  is  far  from  being  constant  (except  over  the  linear  part 
of  the  open-circuit  saturation  curve)  and  must  be  measured  off 
the  diagram  for  each  different  value  of  the  field  excitation.  This 
diagram  (Fig.  125)  shows  very  clearly  the  advantage  of  high  flux 

densities  (magnetic  saturation)  in  some 
portion  of  the  magnetic  circuit,  if  good 
regulation  is  aimed  at. 

)4X(slobs)  107.  Short-circuit  Current.— The 
amount  of  the  short-circuit  current  is  in- 
timately connected  with  the  regulating 
qualities  of  a  machine,  and  in  large  gen- 
erators becomes  a  matter  of  importance. 
The  maximum  value  of  the  armature 
current  at  the  instant  a  short-circuit 
occurs  depends  mainly  on  the  induc- 
tance of  the  armature  windings;  but 

when  the  armature  magnetomotive  force  has  had  time  to  react 
on  the  field  and  has  actually  reduced  the  flux  of  induction  in  the 
air  gap,  the  resulting  current  may  be  fairly  accurately  calculated 
by  using  the  construction  indicated  in  Figs.  127  and  128. 

The  vector  triangle  Fig.  127  is  constructed  for  any  assumed 
value,  Ic,  of  the  armature  current.  It  shows  that  when  the 
terminal  voltage  is  zero,  the  machine  being  short-circuited,  the 
flux  in  the  air  gap  must  be  such  that  the  pressure  OE'g  would  be 
developed  in  the  armature  conductors  on  open  circuit.  The 
value  OF  (Fig.  128)  of  the  ampere-turns  necessary  to  produce  this 
flux  in  the  air  gap  is  thus  obtained,  the  ordinate  OE'g  being  the 
generated  voltage  as  determined  by  the  vector  diagram.  Now 
since  the  magnetomotive  force  of  the  armature  windings  will  be 
almost  wholly  demagnetizing,  it  is  correct  to  assume  that  the  field 
excitation  must  be  increased  by  an  amount  equal  to  the  maxi- 
mum armature  ampere-turns  per  pole  in  order  that  the  resultant 
excitation  may  be  OF.  Thus  FG  in  Fig.  128  is  made  equal  to  the 
maximum  armature  ampere-turns,  and  by  drawing,  to  a  suitable 
scale,  the  ordinate  GJ  equal  to  the  assumed  armature  current  Ic, 


REGULATION  OF  ALTERNATORS 


309 


the  point  /  on  the  short-circuit  current  curve  is  obtained.  By 
repeating  the  construction  for  any  other  assumed  value  of  the 
current  it  will  be  seen  that  so  long  as  E'g  lies  on  the  linear  portion 
of  the  no-load  characteristic,  the  relation  between  the  short- 
circuit  current  and  the  field-pole  excitation  is  also  linear.  When 
the  field  excitation  is  OL,  giving  a  pressure  OE0  on  open  circuit, 
the  short-circuit  current  will  be  LK. 


(SDa 
FIG.  128. — Method  of  constructing  curve  of  armature  current  on  short-circuit. 

108.  Regulation  on  any  Power  Factor. — Unless  the  effects  of 
cross-magnetization  are  taken  into  account,  it  is  impossible  to 
predetermine  the  regulation  accurately  when  the  power  factor 
differs  appreciably  from  zero,  but  by  the  intelligent  use  of  vector 
quantities  (involving  as  they  do  the  assumption  of  simple 
harmonic  curves)  very  satisfactory  results  can  be  obtained.  The 
best  method  known  to  the  author  by  which  the  load  saturation 
curve  for  any  power  factor  may  be  drawn,  without  resorting  to 
flux  distribution  and  wave-shape  analysis,  is  that  given  by 
PROFESSOR  ALEXANDER  GRAY/  and  recently  embodied  in  the 
Standardization  rules  of  the  American  Institute  of  Electrial  En- 
gineers. A.  E.  CLAYTON2  has  also  suggested  a  similar  method. 

*A.  GRAY.     "Electrical  Machine  Design." 
2  Electrician,  vol.  73,  p.  90,  1914. 


310 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


Let  the  external  power  factor  be  cos  6,  and  OR'  (Fig.  129)  the 
constant  field  excitation  which  would,  on  open  circuit,  develop 
the  pressure  E0  represented  by  R'S.  The  full-load-current  zero- 
power-factor  saturation  curve  O'R  has  been  drawn  as  previously 
described.  If  then  it  is  possible  to  determine  the  point  Q  on 
the  full-load  saturation  curve  for  power  factor  cos  6,  the  required 

r\  o 

percentage  regulation  may  be  expressed  as  100^,- 

In  Fig.  130  draw  the  right-angled  triangle  EtPE0  such  that 
PEt  represents  the  armature  resistance  drop  per  phase  with  full- 
load  current,  and  EoP  the  corresponding  synchronous  reactance 
drop,  as  given  by  SR  in  Fig.  129.  From  Et  draw  the  line  Etm 


O  0'  R' 

FIG.  129. — Method   of  constructing  saturation   curve  for  any  load  and 

power  factor. 

of  indefinite  length  and  so  that  mEtP  is  the  required  power-factor 
angle  6.  From  E0  as  center  describe  the  arc  of  a  circle  of  radius 
RfS  (Fig.  129)  equal  to  the  open-circuit  voltage,  and  cutting 
mEt  produced  at  0.  Then  OEt  will  be  the  required  terminal  vol- 
tage, which  may  be  plotted  as  R'Q  in  Fig.  129.  This  construc- 
tion provides  for  the  proper  angle  8  between  terminal  voltage 
and  current;  and  in  regard  to  the  relation  between  the  terminal 
voltage  Et  and  the  open-circuit  voltage  E0  when  load  is  thrown 
off,  it  will  be  seen  that  the  total  synchronous  reactive  drop  has 


REGULATION  OF  ALTERNATORS  311 

been  used  in  the  impedance  triangle  EtPE0  of  I^ig.  130.  This 
virtually  assumes  the  demagnetizing  and  distortional  effects  of 
the  armature  current  to  be  equivalent  to  a  fictitious  reactance 
drop  capable  of  being  treated  vectorially  like  any  other  reactance 
drop,  and  of  which  the  direct  effect  on  regulation  is  proportional 
to  the  sine  of  the  angle  of  lag — a  not  unreasonable  assump- 
tion, though  scientifically  inaccurate.  This  gives  good  results 
in  machines  of  normal  design.  It  is  when  departures  are  made 
from  standard  practice  that  such  approximations  are  liable  to  be 
abused. 


Armature  current 


FIG.  130.— Vector  diagram  showing  construction  to  obtain  terminal  voltage 
for  any  load  and  power  factor. 

109.  Influence  of  Flux  Distribution  on  Regulation. — So  long 
as  a  sinusoidal  air-gap  flux  distribution  can  be  assumed  both  on 
open  circuit  and  under  load  conditions,  the  previously  described 
methods  of  predetermining  regulation  are  satisfactory;  but  in  the 
case  of  new  or  abnormal  designs  of  machines,  correct  results  can 
be  obtained  only  by  taking  into  account  the  alteration  in  the 
amount  of  the  useful  flux  due  to  cross-magnetization  and  the 
changes  in  the  e.m.f.  wave-shapes  due  to  flux  distortion.  An 
attempt  will  be  made  to  outline  as  briefly  as  possible  a  method  of 
study  which,  although  it  has  been  elaborated  by  the  writer,  is  not 
essentially  new;  indeed,  it  is  probably  used  in  a  modified  form  by 
some  practical  designers  when  aiming  at  a  closer  degree  of 
accuracy  than  can  be  expected  from  methods  based  on  the  usual 
sine-wave  assumptions. 

The  method  about  to  be  described  is  based  on  the  fact  that  for 
salient-pole  machines  approximately  correct  flux-distribution 
curves  can  be  drawn  when  the  width  and  shape  of  the  pole  shoe 
have  been  decided  upon;  and  for  high-speed  generators  with  air 
gap  of  constant  length,  when  the  disposition  and  windings  of  the 
slots  in  the  rotor  have  been  determined.  .  From  these  flux  curves, 


312          PRINCIPLES  OF  ELECTRICAL  DESIGN 

whether  representing  open-circuit  or  loaded  conditions,  the  e.m.f . 
waves  and  their  form  factors  can  be  obtained,  all  as  explained  in 
Arts.  100  and  101  of  Chap.  XIII,  and  the  problem  of  regulation 
may  be  summed  up  as  follows :  Plot  the  open-circuit  saturation 
curve  for  the  complete  magnetic  circuit,  correcting  for  the  form 
factor  of  the  developed  voltage  if  this  departs  appreciably  from 
the  assumed  value  of  1.11.  Now  obtain  the  actual  flux  distribu- 
tion and  corresponding  full-load  "apparent"  developed  voltage  for 
a  given  power  factor,  and  correct  for  the  internal  pressure  losses — 
ohmic  and  reactive.  Let  Et  be  full-load  terminal  voltage  obtained 
by  this  method.  The  field  excitation  for  air  gap  and  teeth  is 
known  for  the  particular  condition  considered,  and  the  ampere- 
turns  required  to  overcome  the  reluctance  of  the  remaining  parts 
of  the  magnetic  circuit  are  also  readily  ascertained  since  the 
total  flux  per  pole  (the  area  of  full-load  flux  curve  C)  is  known. 
It  is  therefore  merely  necessary  to  read  off  the  open-circuit 
characteristic  the  voltage  E0  corresponding  to  the  ascertained 
value  of  the  total  field  excitation  in  order  to  determine  the 

T|T     Tjl 

regulation,  which  is     *     — • 
&t 

The  actual  working  out  of  the  problem  is  not  quite  so  simple 
as  this  statement  may  suggest,  the  chief  difficulty  being  that  a 
knowledge  of  the  external  power-factor  angle  is  insufficient  to 
determine  the  exact  position  of  the  armature  m.m.f.  curve  rela- 
tively to  the  center  line  of  the  pole.  The  position  of  this  curve 
depends  upon  the  internal  power-factor  angle  and  also  upon  the 
phase  displacement  of  the  generated  electromotive  force  under 
load  conditions,  i.e.,  on  the  degree  of  distortion  of  the  resultant 
air-gap  flux  which,  on  open  circuit,  was  distributed  symmetric- 
ally about  the  center  line  of  the  pole  face.  The  manner  in  which 
the  displacement  of  the  armature  m.m.f.  curve  may  be  determined 
approximately,  for  any  given  external  power  factor,  was  ex- 
plained in  Art.  98  and  illustrated  by  the  vector  diagram,  Fig.  116. 

110.  Outline  of  Procedure  in  Calculating  Regulation  from 
Study  of  E.m.f.  Waves. — In  Fig.  131  let  the  curve  F  represent  the 
distribution  of  magnetomotive  force  over  the  armature  surface 
tending  to  send  flux  from  pole  to  armature  on  open  circuit. 
Let  BD  be  the  magnetomotive  force  due  to  armature  current 
only.  If  the  load  current  be  sinusoidal  (an  almost  essential  as- 
sumption, since  its  exact  shape  cannot  be  predetermined),  BD 
will  also  be  a  sine  curve,  the  maximum  ordinate  CD  of  which  will 


REGULATION  OF  ALTERNATORS 


313 


be  displaced  beyond  the  center  line  of  the  pole  by  an  amount 
depending  upon  the  power  factor  of  the  load  and  the  distortion 
of  the  resulting  air-gap  flux  distribution.  This  maximum  value 
will  occur  where  the  current  in  the  conductors  is  zero,  and  the 
maximum  armature  current  will  be  carried  by  the  conductor 
displaced  exactly  90  degrees  (electrical  space)  from  the  point  C. 
The  point  B  is  therefore  the  position  on  the  armature  surface, 
considered  relatively  to  the  poles,  where  tlje  current  is  a  maxi- 
mum, the  length  AB  or  0,  which  depends  largely  on  the  power 
factor,  being  determined  approximately  as  explained  in  Art.  98. 
Add  the  ordinates  of  curves  F  and  D  to  get  the  curve  M  which 
gives  the  resultant  magnetomotive  force  under  the  assumed 
conditions  of  load.  Having  calculated  the  permeance  of  the 


FIG.  131. — Distribution  of  m.m.f.  over  armature  surface. 

magnetic  circuit  for  various  points  on  the  armature  surface,  the 
flux  distribution  curves  A0  and  C  of  Fig.  132  can  be  plotted. 
The  first,  which  represents  open-circuit  conditions,  is  plotted 
from  the  m.m.f.  curve  F,  while  curve  C,  showing  the  flux  distribu- 
tion under  load,  is  derived  from  the  m.m.f.  curve  M.  The 
respective  areas  of  these  curves  are  a  measure  of  the  total  air-gap 
flux  under  the  two  conditions,  but  we  cannot  say  that  the  actual 
ampere-turns  on  the  field  will  be  the  same  in  both  cases,  be- 
cause the  component  of  the  total  m.m.f.  required  to  overcome 
the  reluctance  of  the  pole-core  and  yoke  ring  has  not  been 
taken  into  account. 

The   correct   solution   of   the   problem   involves   the   actual 
wave  shapes  of  the  developed  electromotive  forces.     Assuming 


314 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


all  the  flux  in  the  air  gap  to  be  cut  by  the  armature  conductors, 
the  wave  shapes  of  the  " apparent"  developed  e.m.fs.  can  be 
drawn,  and  their  form  factors  calculated  as  explained  in  Arts.  100 
and  101.  The  terminal  voltage — which  must  be  known  before 
the  regulation  can  be  calculated — is  most  readily  obtained  by 
using  vector  diagrams;  but  this  involves  the  substitution  of 
"  equivalent  sine  curves  "  for  the  irregular  waves.  The  maximum 
value  of  a  so-called  equivalent  sine  wave  is  \/2  times  the  r.m.s. 
value  of  the  irregular  wave;  but  its  time  phase  relatively  to  any 
defined  instantaneous  value  of  the  irregular  wave  is  not  so  easily 
determined.  It  can  be  obtained  from  the  irregular  curve  when 
plotted  to  polar  coordinates  as  explained  in  Art.  102,  Chap.  XIII ; 


FIG.  132. — Flux    distribution,    (C)    under   load,   and   (A0)   when    load  is 

thrown  off. 

but  a  method  to  be  preferred  for  purposes  of  explanation,  although 
more  tedious,  consists  in  obtaining  the  average  value  of  the  true 
power  and  making  the  displacement  between  electromotive  force 

/     true  power     \          , 

and   current   vectors  equal   to   cos"1    I-  — ).     The 

\apparent  power/ 

current  wave  (assumed  to  be  a  sine  curve)  from  which  the  m.m.f . 
curve  BD  of  Fig.  131  is  derived  would  have  its  maximum  value 
at  the  point  B,  displaced  /3  electrical  degrees  beyond  the  center, 
A,  of  the  pole.  The  actual  full-load  e.m.f.  wave,  can  also  be 
drawn  in  the  correct  position  relatively  to  the  center  line  of  the 
pole;  and,  by  multiplying  the  corresponding  instantaneous 
values  of  electromotive  force  and  current,  the  power  curve  can  be 
drawn  and  the  average  value  of  its  or dinat es  calculated .  The  ratio 


REGULATION  OF  ALTERNATORS  315 

of  this  quantity  to  the  volt-amperes  is  equal  to  the  cosine  of  the 
angle  \j/'  in  Fig.  133.  This  vector  diagram  can  be  constructed  as 
follows : 

Draw  OE0  representing  the  phase  of  the  open-circuit  voltage, 
i.  e.,  the  center  of  the  pole,  to  be  used  as  a  reference  line 
from  which  the  phase  angles  can  be  plotted.  Make  the  angle 
E0OIC  equal  to  ft  of  Fig.  131.  This  is  the  estimated  lag  of 
current  behind  the  open-circuit  electromotive  force.  Draw 
OE'a  equal  in  length  to  the  calculated  e.m.f.  value  of  the  "ap- 
parent" developed  voltage  under  load  conditions,  and  so  that  \f/f  = 

I       watts       \ 

cos"  1 1  —  ) ,  where  the  watts  referred  to  are  calculated 

\  volt-amperes/ 

by  multiplying  the  corresponding  instantaneous  values  of  E'g 
and  Ic.  From  E'g  drop  a  perpendicular  on  to  0/c,  and  set  off 


IX  (  slots) 
Eg 
/Abends) 


FIQ.  133 — Vector  diagram  for  determining   the  inherent  regulation  of  an 
alternating-current  generator. 

E'g  E0  and  EgP  to  represent  the  reactance  drops  per  phase  in  slots 
and  end  connections  respectively.  Draw  PEt  parallel  to  OIe  to 
represent  the  resistance  pressure  drop  per  phase,  and  join  the 
point  0  with  Eg  and  Et  respectively.  The  angle#(0/c  and  the 
length  of  the  vector  OEt  may  not  correspond  with  the  exact 
values  of  external  power  factor  and  terminal  voltage  assumed 
when  the  angle  ft  was  originally  estimated;  but,  by  using  the 
vector  construction  on  the  assumption  of  sine-waves  throughout, 
a  very  close  estimate  of  these  quantities  can  be  made.  The  im- 
portant point  in  connection  with  this  method  of  analysis  is  that 
the  external  power-factor  angle  6  and  the  terminal  voltage  Et  can 
be  calculated  for  any  value  of  the  armature  current  7C  when  the 
phase  displacement  of  the  latter  relatively  to  the  open-circuit 
voltage  is  assumed. 


316  PRINCIPLES  OF  ELECTRICAL  DESIGN 

The  field  ampere  turns  necessary  to  produce  the  terminal 
voltage  OEt  of  the  vector  diagram  Fig.  133  are  made  up  of  the 
ampere  turns  for  air  gap  and  armature  teeth,  represented  by  the 
maximum  ordinate  of  the  curve  F  of  Fig.  131,  together  with  the 
ampere  turns  required  to  overcome  the  reluctance  of  the  pole- 
core,  yoke  ring,  and  armature  core.  These  additional  ampere 
turns  are  readily  calculated  because  the  total  useful  flux  per 
pole  is  known,  being  represented  by  the  area  of  the  curve  C  of 
Fig.  132. 

Having  determined  the  total  ampere  turns  per  pole  which  are 
necessary  to  give  OEt  volts  (of  Fig.  133)  at  the  terminals,  it  is 
easy  to  read  the  corresponding  open-circuit  voltage  from  the 
no-load  saturation  curve  of  the  machine.  In  this  manner  the 
regulation  corresponding  to  a  known  external  power  factor,  cos 
6,  can  be  calculated  with  greater  accuracy  than  will  usually  be 
obtained  by  the  method  outlined  in  Art.  108,  and  illustrated 
by  Figs.  129  and  130. 

The  meaning  of  the  other  quantities  in  Fig.  133  may  be  summed 
up  as  follows: 

The  angle  E0OE'g,  or  a,  is  the  phase  difference  between  equiva- 
lent sine-waves  representing  open-circuit  voltage  and  " apparent" 
developed  voltage  under  load  conditions.  It  is  the  result  of 
flux  distortion  due  to  the  armature  cross-magnetizing  ampere- 
turns.  The  vector  OEg  gives  the  r.m.s.  value  of  the  voltage  per 
phase  actually  developed  in  the  armature  winding  by  the  cutting 
of  the  flux  linking  with  the  " active"  conductors. 

The  angle  EgOIc  or  \f/  is  the  internal  power-factor  angle.  The 
difference  in  length  between  OE0  and  OE'g  is  the  voltage  drop 
due  to  armature  demagnetization  and  distortion.  The  point 
E0  is  shown  in  Fig.  133  on  PE'g  produced,  but  it  does  not  neces- 
sarily fall  on  this  straight  line,  and  so  indicates  one  important 
difference  between  the  construction  of  Fig.  133  and  that  of 
Fig.  130,  in  which  the  assumptions  made  are  not  universally 
applicable. 

The  use  of  vectors  and  vector  constructions,  such  as  were  first 
described,  will  usually  give  sufficiently  accurate  results  without 
the  expenditure  of  time  and  labor  involved  in  the  plotting  of 
flux  curves  and  e.m.f.  waves.  It  is  in  the  case  of  abnormal 
designs,  or  when  the  conditions  are  unusual,  that  the  problem 
of  regulation  may  be  studied  most  conveniently  and  correctly 
by  a  method  such  as  that  here  described,  which  is  subject 


REGULATION  OF  ALTERNATORS 


317 


to  modification  in  matters  of  detail  and  may  be  elaborated  if 
desired. 

In  the  writer's  opinion,  a  further  advantage  of  the  method  of 
flux  distribution  and  wave-form  analysis  lies  in  the  fact  that  the 
designer  obtains  thereby  a  clearer  conception  of  the  factors  enter- 
ing into  the  problem  of  regulation  than  he  is  ever  likely  to  obtain 
if  he  confines  himself  to  the  use  of  formulas  and  vector  diagrams, 
which  are  always  liable  to  be  abused  when  familiarity  with  their 
purpose  and  construction  leads  to  forgetfulness  of  their  meaning 
and  limitations. 

111.  Efficiency. — In  estimating  the  efficiency  of  an  alternating- 
current  generator  before  it  is  built,  the  same  difficulties  occur  as 
in  the  case  of  the  dynamo.  There  are  always  some  losses  such 
as  windage,  bearing  friction,  and  eddy  currents,  which  cannot 
easily  be  predetermined,  and  it  is  therefore  necessary  to  include 
approximate  values  for  these  in  arriving  at  a  figure  for  the  total 
losses.  Very  little  need  be  added  to  what  has  already  been  said 
in  Art.  60  of  Chap.  IX,  to  which  the  reader  is  referred.  He  should 
also  consult  the  working  out  of  the  numerical  example  under 
items  (148)  and  (149)  in  Art.  63;  and  make  a  list  of  all  the  losses 
occurring  in  the  machine  at  the  required  output  and  power  factor. 

In  the  ratio  efficiency  = 


output 

' li  ls  the  actual  output  of 


the  generator  at  a  given  power  factor  with  which  we  are  con- 
cerned, and  not  the  rated  k.v.a.  output. 

Windage  and  bearing  friction  losses  are  never  easily  estimated ; 
but  the  following  figures  may  be  used  in  the  absence  of  more 
reliable  data. 


APPROXIMATE  WINDAGE  AND  FRICTION  LOSSES  EXPRESSED  AS  PERCENTAGE 
OF  FULL-LOAD  OUTPUT 

K.v.a.  output 

f  50 

Self -ventilated  A.-C.  generators \         200 

500  and 
larger 


Turbo-alternators:   forced  ventilation 
(exclusive  of  power  to  drive  fan) .... 


2,000 

5,000 

10,000 

15,000 

20,000 


Windage  and 
bearing  friction 

1.5  per  cent. 
1.0  per  cent. 

0.5  per  cent. 

1 . 8  per  cent. 
1 . 5  per  cent. 
1 . 2  per  cent. 
1.0  per  cent. 
0 . 9  per  cent. 


318          PRINCIPLES  OF  ELECTRICAL  DESIGN 

The  power  required  to  drive  the  ventilating  fan  for  turbo- 
alternators  will  generally  be  from  0.3  to  0.5  per  cent,  of  the  rated 
full-load  output  of  the  generator. 

The  brush-friction  loss  is  usually  small.  If  A  is  the  total  area 
of  contact,  in  square  inches,  between  brushes  and  slip  rings,  and 
v  is  the  peripheral  velocity  of  the  slip  rings  in  feet  per  minute, 

vA 
the  brush-friction  loss  will  be  approximately   TT  watts. 


The  hysteresis  and  eddy-current  losses  in  the  iron  can  be  cal- 
culated for  any  given  load  because  the  required  developed  voltage, 
and  therefore  the  total  flux  per  pole,  are  known.  The  losses  in 
the  teeth  can  now  be  calculated  with  greater  accuracy  than 
before  the  full-load  flux  curves  were  drawn,  because  the  maximum 
value  of  the  tooth  density  will  depend  upon  the  maximum  value 
of  the  air-gap  flux  density  as  obtained  from  the  flux  distribution 
curve  for  the  loaded  machine,  as  explained  in  Art.  60  (page 
196).  Unless  the  density  in  the  teeth  is  high,  it  is  usually  un- 
necessary to  calculate  the  actual  tooth  density  because  the 
"apparent"  tooth  density  may  be  used  in  estimating  tooth  losses; 
but  with  low  frequency  machines  the  density  in  the  teeth  may 
be  carried  well  above  the  "knee"  of  the  B-H  curve,  and  it  would 
then  be  necessary  to  determine  the  actual  tooth  density  as  ex- 
plained in  Art.  37  of  Chap.  VII  under  dynamo  design. 

Eddy  Currents  in  Armature  Conductors.  —  The  PR  loss  in  the 
armature  copper  may  be  calculated  when  the  cross-section  and 
length  of  the  winding  are  known  ;  but  in  the  case  of  large  machines 
with  heavy  conductors,  the  eddy-current  loss  in  the  "active" 
conductors  may  be  considerable,  and  an  allowance  should  then 
be  made  to  cover  this.  The  eddy  currents  in  the  buried  portions 
of  the  winding  are  due  to  two  causes: 

1.  The  flux  entering  the  sides  of  the  teeth  through  the  top  of 
the  slot. 

2.  The  slot  leakage  flux  which  the  armature  conductors  them- 
selves produce  when  the  machine  is  delivering  current  to  the 
circuit. 

The  loss  due  to  (1)  is  independent  of  the  load,  and  would  be  of 
importance  in  the  case  of  solid  conductors  of  large  cross-section 
in  wide  open  slots.  With  narrow,  or  partially  closed,  slots,  it  is 
negligible;  but  occasion  arises  when  it  is  advisable  to  laminate 
the  conductors  in  the  upper  part  of  the  slot  to  avoid  appreciable 
loss  due  to  this  cause. 


REGULATION  OF  ALTERNATORS  319 

Item  (2)  may  lead  to  very  great  additional  copper  losses  if 
solid  conductors  of  large  cross-section  are  used  in  narrow  slots 
of  considerable  depth.  The  calculation  of  the  losses  due  to  the 
reversals  of  the  slot  leakage  flux  could  be  made  without  difficulty 
if  it  were  not  for  the  fact  that  the  dampening  effect  of  the  unequal 
distribution  of  the  current  density  through  the  section  of  the 
solid  conductor  actually  decreases  the  amount  of  the  slot  flux 
and  so  reduces  the  loss.  The  best  and  most  thorough  treatment 
of  this  subject  known  to  the  writer  is  that  of  PROF.  A.  B.  FIELD 
in  the  Trans.,  A.  I.  E.  E.,  vol.  24,  p.  761  (1905).  The  remedy  in 
the  case  of  heavy  losses  due  to  slot  leakage  flux  through  the  cop- 
per is  to  laminate  the  conductors  in  a  direction  parallel  to  the 
flux;  thus,  if  copper  strip  is  used,  it  must  not  be  placed  on  edge 
in  the  slot,  but  should  be  laid  flat  with  the  thin  edge  presented 
to  the  flux  lines  crossing  the  slot,  exactly  as  in  the  case  of  the 
armature  stampings,  which  are  so  placed  relatively  to  the  flux 
from  the  poles.  Owing  to  the  fact  that  the  leakage  flux  is  con- 
siderably greater  in  amount  near  the  top  than  the  bottom  of  the 
slot,  the  losses  due  to  both  causes  of  flux  reversal  in  the  space 
occupied  by  the  "active"  copper  are  of  more  importance  in  the 
upper  layers  of  conductors  than  in  those  near  the  bottom  of  the 
slot.  For  this  reason,  the  upper  conductors  will  sometimes  be 
laminated,  while  the  lower  conductors  are  left  solid.  When  the 
method  of  lamination  has  been  decided  upon,  the  probable  in- 
crease in  loss  can  be  obtained  from  figures  and  curves  published 
by  PROF.  FIELD  in  the  paper  previously  referred  to;  but  it  is 
suggested  that,  for  the  purpose  of  estimating  the  probable  effi- 
ciency, the  calculated  PR  loss  in  the  armature  windings  be  in- 
creased 15  per  cent,  in  the  case  of  slow-speed  machines  of  moder- 
ate size,  and  30  per  cent,  in  the  case  of  steam-turbine-driven 
units  of  large  output.  This  addition  is  intended  to  cover  not 
only  the  losses  due  to  eddy  currents  in  the  armature  windings, 
but  all  indeterminate  losses  in  end  plates,  supporting  rings,  etc., 
which  increase  with  the  load. 


CHAPTER  XV 
EXAMPLE  OF  ALTERNATOR  DESIGN 

112.  Introductory. — The  principles  and  features  of  alternator 
design,  as  given  in  the  foregoing  chapters,  will  now  be  applied 
and  illustrated  in  the  working  out  of  a  numerical  example.  A 
steam-turbine-driven  three-phase  generator  will  be  selected,  be- 
cause this  design  involves  greater  departures  from  the  previously 
illustrated  D.C.  design  than  would  occur  if  the  slow-speed  type 
of  alternator  with  salient  poles  were  selected.  It  is  true  that  the 
difficulties  encountered  in  the  design  of  turbo-alternators — espe- 
cially of  the  larger  sizes,  running  at  exceptionally  high  speeds — 
are  of  a  mechanical  rather  than  an  electrical  nature;  but  this 
merely  emphasizes  the  importance  to  the  electrical  engineer  of  a 
thorough  training  in  the  principles  and  practice  of  mechanical 
engineering. 

It  is  not  possible  for  a  man  who  is  not  in  the  first  place  an  ex- 
perienced mechanical  engineer  to  design  successfully  a  modern 
high-speed  turbo-alternator.  These  machines  are  now  made  up 
to  30,000  k.v.a.  output  at  1,500  revolutions  per  minute  (25  cycles) 
and  35,000  k.v.a.  at  1,200  revolutions  per  minute  (60  cycles). 
Larger  units  can  be  provided  as  the  demand  arises;  it  is  probable 
that  single  units  for  outputs  up  to  50,000  k.v.a.  at  750  revolutions 
per  minute  will  be  built  in  the  near  future.  With  the  great 
weight  of  the  slotted  rotors,  carrying  insulated  exciting  coils,  and 
travelling  at  very  high  peripheral  velocities,  new  problems  have 
arisen,  and  these  problems  should  be  seriously  studied  by  anyone 
proposing  to  take  up  the  design  of  modern  electrical  machinery. 
Engineering  textbooks  may  constitute  a  basis  of  necessary 
knowledge;  but,  with  the  rapid  advance  in  this  field  of  electrical 
engineering,  the  information  of  greatest  value  (apart  from  what 
the  manufacturing  firms  deliberately  withhold)  is  to  be  found  in 
current  periodical  publications,  including  the  papers  and  discus- 
sions appearing  in  the  journals  of  the  engineering  societies. 

Since  it  will  not  be  possible  to  discuss  the  mechanical  details 
of  turbo-alternator  designs  in  these  pages,  a  machine  of  medium 
size  (8,000  k.v.a.)  will  be  chosen,  and  the  peripheral  speed  of 
the  rotor  will  not  be  permitted  to  exceed  18,000  ft.  per  minute. 
The  mechanical  difficulties  will  therefore  not  be  so  great  as  in 

320 


EXAMPLE  OF  ALTERNATOR  DESIGN  321 

some  of  the  larger  machines  running  at  higher  peripheral  veloci- 
ties, and  the  electrical  features  of  the  design  will  be  considered 
alone,  reference  being  made  to  mechanical  details  only  as  occa- 
sion may  arise. 

It  is  proposed  to  work  through  the  consecutive  items  of  a  design 
sheet,  as  was  done  for  the  D.C.  dynamo;  but  the  sheets  will  include 
fewer  detailed  items,  more  latitude  being  allowed  in  the  exercise 
of  judgment  and  the  application  of  knowledge  derived  from  the 
work  done  on  previous  designs.  An  attempt  will  be  made  to 
render  the  example  of  use  in  the  design  of  slow-speed  salient-pole 
machines,  and,  with  this  end  in  view,  references  will  be  made  to 
the  text  when  taking  up  in  detail  the  items  of  the  design  sheet. 
For  the  same  reason — namely,  to  make  the  numerical  example  of 
broad  application — the  writer  may  take  the  liberty  of  digressing 
sometimes  from  the  immediate  subject,  if  matters  of  interest 
suggest  themselves  as  the  work  proceeds. 

113.  Single-phase  Alternators. — Since  the  selected  design  is 
that  of  a  polyphase  machine,  it  seems  advisable  to  state  here  one 
or  two  matters  of  special  interest  in  the  design  of  the  less  common 
single-phase  generator.  It  is  easier  to  design  a  polyphase  than 
a  single-phase  alternator,  although  this  fact  is  not  always  recog- 
nized, even  by  designers.  Many  of  the  single-phase  machines 
in  actual  service  are  less  efficient  than  they  might  be;  but  the 
problems  which  are  peculiar  to  single-phase  generators  receive 
comparatively  little  attention  because  these  machines  are  rarely 
used  at  the  present  time,  the  development  during  recent  years 
having  been  mainly  in  the  direction  of  power  transmission  and 
distribution  by  polyphase  currents. 

It  is  the  pulsating  nature  of  the  armature  m.m.f.,  as  explained 
in  Art.  94,  Chap.  XIII  that  leads  to  eddy-current  losses  that 
are  practically  inappreciable  in  the  case  of  two-  or  three-phase 
machines  working  on  a  balanced  load,  that  is  to  say,  with  the 
same  current  and  the  same  voltage  in  each  of  the  phase  windings, 
and  with  the  same  angular  displacement  between  current  and 
e.m.f.  in  the  respective  armature  circuits.  Sometimes  the  poly- 
phase load  is  not  balanced,  and  in  that  case  pulsations  of  the 
armature  field  occur  as  in  the  single-phase  machines,  the  amount 
of  the  pulsating  field  being  dependent  upon  the  degree  of  un- 
balancing of  the  load.  The  effect  is  then  as  if  an  alternating 
field  were  superposed  on  the  steady  armature  m.m.f.  due  to 
the  balanced  components  of  the  total  armature  current. 

Without  going  into  detailed  calculations,  it  may  be  stated  that 
21 


322 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


the  remedy  consists  in  providing  ammortisseur — or  damper- 
windings  on  the  pole  faces,  as  mentioned  in  Art.  94,  Chap.  XIII. 
A  simple  form  of  damper  is  illustrated  in  Fig.  134,  where  a  num- 
ber of  copper  rods  through  the  pole  face — and  on  each  side  of 


FIG.  134. — Short-circuited  damping  bars  in  pole  face  of  single-phase  alternator. 

the  pole  shoe — are  short-circuited  at  both  ends  by  heavy  bands 
of  copper.  Any  tendency  to  sudden  or  periodic  changes  of  flux 
through  any  portion  of  the  pole  face  is  checked  to  a  very  great 
extent  by  the  heavy  currents  that  a  small  change  of  flux  will 
establish  in  the  short-circuited  rods. 

114. — Design  Sheets  for  Alternating-current  Generator. — 


SPECIFICATION 


Output,  k.v.a 

Number  of  phases. .  . 
Terminal  voltage  .  .  . 
Power  factor  of  load . 

Frequency 

Type  of  drive 

Speed,  r.p.m 

Inherent  regulation . 


8,000 

3 

6,600 

0.8 
60 

Steam  turbine 
1,800 
Within  25  per  cent,  rise  when  full 

load  is  thrown  off. 
130 


Exciting  voltage 

Permissible   temperature   rise   after  6  hr.  full-load   run 

(by  thermometer) 45°C. 

Ventilating  fan independently  driven    (not  part 

of  generator). 

GENERAL  OUTLINE  OP  PROCEDURE 
(a)  Design  armature. 
(6)   Design  field  magnets. 

(c)  Draw  flux  distribution  curves.     Obtain  wave  shapes  and  form  factors. 

(d)  Complete  field  system.     Open-circuit  saturation  curve.     Regulation,  and  short-circuit 
current. 

(e)  Efficiency. 


EXAMPLE  OF  ALTERNATOR  DESIGN 


323 


CALCULATIONS 


Sym- 
bols 

Assumed  or 
approximate 
values 

Final 
values 

1     Number  of  poles 

P 

r 

/ 

Q 
la 

18.000 
38.2 

4 

38.25 
700 
Y 
800 
700 
11,340 
H 
40 
31.416 
inding 
48 
4 
2.62 
3 
62.2X10* 
5,910 
51 
46.8 
49H 
layer 
2,000 

4X9*X0.14 

1 
4M 

14,300 
8,500 
14 
28,000 
4,900 
120  kw. 
210 
0.01135 
8.4 
21 
116 
190 
3,875 
65.2X10* 
drical) 
8 
3.76 
1.625 
0.3059 
1.25 

122.3 

2.   Peripheral  speed  (feet  per  minute)  .  .  . 
3.   Diameter  of  rotor  (inches)  
4    Line  current 

5.   Phase  connection  (star  or  delta)  
6    Specific  loading 

Y 

812 

8.  Armature  ampere-turns  per  pole... 
9.  Length  of  air  gap  at  center  (inches)..  . 
10.   Diameter  of  stator  (armature)  (inches)  
11.   Pole  pitch  (inches)  
12.  Pole  arc  (inches)  
13.   Number  of  inductors  per  phase  
14.   Number  of  armature  slots  per  pole  per  phase.  . 
15.  Slot  pitch 

(S/)« 

a 

D 

r 
] 

z 

n 
\ 
T. 
* 
Bg 
I. 

In 

12,7.50 

% 
40 

distributed  w 
48.7 
4 
2.62 
3 

6.000 
51 

16.   Number  of  inductors  per  slot 

17.  Flux  per  pole  (no  load) 

18.  Average  flux  density  over  pole  pitch  (open  circuit)... 

19.  Axial  length  of  armature  core  (  «"•?.  (in.ch">)  '  ' 
\  net  (inches)  .  .  . 

20    Axial  length  of  pole  face  (inches) 

21.  Determine  style  of  winding  
22.  Current  density  in  armature  conductors  
23.  Size  of  conductor.     How  made  up.     Slot  insulation 

A 

Single 
2,000 

24.  Tooth  and  slot  proportions. 
25.  Width  of  armature  slot  (inches)  
26.   Depth  of  armature  slot  (inches)                   

« 
d 

27.  Apparent  tooth  density  (no  load)  at  center  of  tooth 
28.  Flux  density  in  armature  core 

29.  Radial  depth  of  armature  core  below  teeth  (inches)  . 
30.  Weight  of  core  (iron)  (pounds) 

Rd 

31    Weight  of  teeth  (iron)  (pounds) 

32.  Total  core  loss,  including  teeth  (open  circuit)  
33    Length  mean  turn  of  armature  coils  (inches) 

34    Resistance  per  phase  (ohms) 

35    IR  drop  per  phase  (full-load  current) 

volts 

EgP 
E'0E0 

36.  Total  armature  copper  loss  (full-load  current)  kw  .  .  . 
37.  IX  drop  (ends)  —  volts  per  phase  winding  
38.  IX  drop  (slots)  —  volts  per  phase  winding    
39.  Full-load  developed  voltage  (per  phase  winding)    .  .  . 
40.  Full-load  flux  per  pole  
41.  Shaping  of  pole  face  

3,875 
65.2X10" 
(Cylin 
8 
3.76 
1.625 

42.  Number  of  slots  per  pole  (rotor)    
43    Slot  pitch  (rotor) 

44.  Slot  width  (rotor)  

45.  Permeance  per  square  centimeter  of  air  gap  (center) 

47.  "Actual"  tooth  density  in  terms  of  air-gap  density 
48.  Saturation  curves  for  air  gap,  teeth,  and  slots  
49.  Permeance  curve  (if  salient-pole  design). 
50.  Open-circuit  flux  curve  A  
51.  M.m.f.  curve  for  flux  curve  A  
52.  Required  area  of  full-load  flux  curve  C  

Fig.   137 
Fig.  139 

Fig.  141 
Fig.   142 

324 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


CALCULATIONS. — Continued 


Sym- 
bols 

Assumed  or 
approximate 
values 

Final 
values 

53.  Resultant  m.m.f  .  for  flux  curve  C  
54.  Full-load  flux  curve  C  
55.  E.m.f.  wave  shapes  at  no  load  and  full  load  
56.  Form  factor,  no  load. 

Fig.   1 
Fig.  1 
Figs. 

42 
41 
L43,  144 

1    10 

58.  Complete  field-magnet  design. 

If 

1   15 

60.  Cross-section  of  pole  cores. 
61.  Cross-section  of  yoke  ring. 
62.  Open-circuit  saturation  curve  for  complete  magnetic 
circuit  

Fig.  1 

46 

27,000 

37,000 

1HXO  12 

375 

514 

68    I^R  loss  (field)  ;  full  load  

66  kw. 

69.  Total  cross-section  of  air  ducts  (forced  ventilation)  .  . 

.,  .,    f  Unity  power  factor.  .  . 
70.  Inherent     regulation     (full-      g            d              factor 

load  current)        .                   {  Zero  power  factor  .... 
71.  Short-circuit  current  (per  phase  winding)  with  full- 
load  excitation  (amperes)  
72.  Efficiency  at  full  load  and  specified  power  factor.  .  .  . 
73.  Efficiency  at  fractional  loads. 
74.  Approximate  volume  of  air  required  (forced  ventila- 
tion) —  cubic  feet  per  minute         

v 

6  .  6  sq.  ft. 
22  per  cent. 

1,950 
0.955 

29,000 

75.  Average  velocity  of  air  in  ventilating  ducts  —  feet  per 

4,400 

115.  Numerical  Example. — Calculations. — Items  (I)  to  (11). 
With  a  frequency  of  60  cycles  per  second  and  a  speed  of  1,800 
revolutions  per  minute,  the  number  of  poles  is 

2  X  60  X  60 


At  a  peripheral  velocity  of  18,000  ft.  per  minute,  the  diameter 

.  .'  , ,  ,     18,000  X  12       00     . 

of  the  rotor  would  be    .  gnn  ^ —  =  38.2  in. 

J-,oUU    /x  7T 

Since  the  air  gap  is  not  likely  to  be  less  than  1  in.,  let  us  decide 
upon  an  internal  diameter  of  the  armature  D  =  40  in.  (item  (10)), 
and  determine  the  exact  dimensions  of  the  rotor  after  the  air 
gap  has  been  decided  upon. 

The  line  current  (item  (4))  is 

800,000 


\/3  X  6,600 


700  amp., 


'     EXAMPLE  OF  ALTERNATOR  DESIGN  325 

and  if  we  select  the  star  connection  of  phases,  this  is  also  the 
current  per  phase  winding  (item  (7)). 

Referring  to  Art.  75  for  values  of  the  specific  loading  (page  250) 
we  find  that  the  average  value  there  suggested  is  q  =  650;  but 
the  peripheral  loading  is  greater  in  turbo-alternators  than  in 
slow-speed  salient-pole  machines,  because  it  is  desired  to  keep 
the  axial  length  as  short  as  possible,  and  the  greater  losses  per 
unit  area  of  cooling  surface  can  be  dealt  with  by  a  suitable  system 
of  forced  ventilation.  We  may  therefore  select  a  value  for  q 
equal  to,  or  even  greater  than,  the  proposed  maximum  for  self- 
cooled  machines.  Let  us  try  q  =  650  X  1.25  =  812. 

The  pole  pitch  (item  (11))  is 

TT  X40 
T  =    -  v—  =  31.416  in. 

and  the  approximate  armature  ampere-turns  per  pole  (item  (8)) 
will  be 

31.42X812 
(SI)  a  =        — 2-         =  12,750 

Referring  to  Arts.  76  and  77,  Chap.  XI,  we  can  get  a  prelimi- 
nary idea  of  the  required  length  of  air  gap  as  explained  in  Art.  77. 
We  shall,  in  this  design,  deliberately  select  a  high  value  for  the 
air-gap  flux  density,  and  if  necessary  saturate  the  teeth  of  the 
rotor  while  keeping  the  density  in  the  armature  teeth  within  rea- 
sonable limits  to  prevent  excessive  hysteresis  and  eddy-current 
loss.  Let  us  try  Bg  =  6,000  gausses,  which  is  higher  than  the 
upper  limit  of  the  range  suggested  in  Art.  76.  The  principal 
advantage  of  using  high  flux  densities  is  that  the  axial  length  of 
the  rotor  can  thus  be  reduced;  but  if  it  is  found  later  that  the 
selected  value  of  Bg  leads  to  unduly  high  flux  density  in  the  stator 
teeth,  it  will  have  to  be  modified. 

The  probable  maximum  value  of  the  air-gap  density  on  open 

circuit  is  o  X  6,000  =  9,450  gausses;  and,  assuming  the  ratio  of 
m.m.fs.  to  be  1.25,  we  have 

9,450  X  5  X  2.54  =  1.25  X  0.4^  X  12,750 

whence  d  =  0.835  or  (say)  %  in- 

In  the  case  of  medium-speed  salient-pole  designs,  the  periph- 
eral velocity  would  not  be  decided  upon  by  merely  selecting 
the  upper  limit  of  8,000  ft.  per  minute  as  given  in  Art.  66  (page 


326          PRINCIPLES  OF  ELECTRICAL  DESIGN 

238) .  This  has  to  be  considered  in  connection  with  the  pole  pitch 
(see  Art.  74),  a  preliminary  diameter  of  rotor  being  selected  in 
keeping  with  what  seems  to  be  a  reasonable  pole  pitch.  A  few 
rough  calculations  will  very  soon  show  whether  or  not  the  tenta- 
tive .value  of  T  will  lead  to  a  suitable  axial  length  of  armature 
core. 

Items  (13)  to  (16). — On  the  basis  of  q  =  812,  the  number  of 
conductors  per  phase  would  be 

Z  =  l(^}   =  48.7 


With  four  slots  per   pole  per  phase,  and  three  conductors  in 
each  slot,  we  have  Z  =  3X4X4=48. 

For  item  (15)  we  have  X  =  — ^  -  =  2.62  in.,  giving  a  cor- 

\.£t 

700  X  3 

rected  value  for  peripheral  loading  of  q  =  — ^TT~  =  800  approx. 

z.oz 

Items  (17)  to  (20). — For  the  purpose  of  calculating  the  flux 
required  on  open  circuit,  we  may  use  formula  (94)  of  Art.  70, 


/>  i 
where  Eper  phase  =    '  /-,  and  k  =  0.958. 

The  required  flux  per  pole  is  therefore 

6,600  X  108 


V3  X  2.22  X  0.958  X  60  X  48 
=  62.2  X  106  maxwells. 

With  the  assumed  value  of  6,000  gausses  for  Bg,  the  axial 
length  of  armature  core  will  be 

62.2  X  106 
a  ~  6,000X6.45X31.42  " 

This  is  a  short  armature  for  a  machine  with  a  rotor  38.25  in. 
in  diameter;  but  it  is  what  we  are  aiming  at,  and  if  the  field 
winding  can  be  accommodated  in  the  space  available,  the  design 
should  be  satisfactory. 

We  shall  attempt  to  ventilate  this  generator  by  means  of  axial 
air  ducts  only.  If,  then,  there  are  no  radial  air  spaces,  the  net 
length  of  iron  in  the  armature  core  will  be  approximately 
ln  =  0.92Za  =  46.8  in.  (Art.  84);  but  these  dimensions  cannot 
be  finally  decided  upon  until  the  slot  proportions  and  tooth 
densities  have  been  settled. 

Whirling  Speed  of  Rotor. — A  matter  of   considerable  impor- 


EXAMPLE  OF  ALTERNATOR  DESIGN  327 

tance  in  the  design  of  high-speed  machinery  is  the  particular 
speed  at  which  vibration  becomes  excessive.  Without  attempt- 
ing to  go  into  the  mechanical  design  of  shaft  and  bearings,  it 
should  be  pointed  out  that  the  size  of  shaft  in  turbo-alternators 
is  determined  mainly  by  what  is  known  as  the  whirling  speed, 
which,  in  turn,  depends  upon  the  deflection  of  the  rotor  considered 
as  a  beam  with  the  points  of  support  at  the  centers  of  the  two 
bearings.  There  will  be  one  or  more  critical  speeds  at  which 
the  frequency  of  the- bending  due  to  the  weight  of  the  rotating 
part  will  correspond  exactly  with  the  natural  frequency  of  vi- 
bration of  the  shaft  considered  as  a  deflected  spring.  The  vibra- 
tion will  then  be  excessive,  causing  chattering  in  the  bearings 
and  abnormal  stresses  which  may  lead  to  fracture  of  the  shaft. 

The  maximum  deflection  of  the  rotor  due  to  its  own  weight 
together  with  the  unbalanced  magnetic  pull  (if  any)  can  be 
calculated  within  a  fair  degree  of  accuracy  when  the  position  of 
the  bearings  and  the  cross-section  of  the  shaft  are  known.  The 
whirling  speed  of  a  rotor  with  steel  shaft,  in  revolutions  per  min- 
ute, can  then  be  calculated,  because  it  is  approximately 

190 


V  Deflection  in  inches 

In  turbo-alternators  the  whirling  speed  is  generally  higher 
than  the  running  speed;  but  this  is  not  a  necessary  condition 
of  design;  and  in  direct-current  steam-turbine-driven  dynamos, 
where  the  provision  of  a  commutator  calls  for  the  smallest  possible 
diameter  of  shaft,  the  whirling  speed  is  commonly  lower  than  the 
normal  running  speed.  In  such  cases  it  is  necessary  to  pass 
through  the  critical  speed,  causing  vibration  of  the  rotor,  every 
time  the  machine  is  started  or  stopped;  but  this  is  not  a  serious 
objection.  A  good  rule  is  to  arrange  for  the  whirling  speed  to 
be  either  25  per  cent,  above,  or  25  per  cent,  below,  the  running 
speed.  Taking  as  an  example  the  design  under  consideration, 
the  whirling  speed  should  be  either  1,800  +  25  per  cent.  =  2,250; 
or  1,800  —  25  per  cent.  =  1,350.  In  the  first  place  the  permis- 
sible deflection  would  be  (o^n)  =  0.00714  in.;  and  in  the 

(190  \  2 
J-OKQ)     =  0.0198  in. 

By  making  a  very  rough  estimate  of  the  rotor  weight  and  the 
span  between  bearings,  it  will  be  seen  that  the  smaller  deflection 


328          PRINCIPLES  OF  ELECTRICAL  DESIGN 

(corresponding  to  the  higher  critical  speed)  is  easily  attainable, 
the  diameter  of  the  shaft  being  of  the  order  of  13  in.  near  the 
rotor  body,  and  10^  in.  in  the  bearings.  It  is  not  proposed  to 
go  further  into  details  of  mechanical  design;  but  attention  may 
be  called  to  the  fact  that  a  rotor  forged  solid  with  the  shaft, 
or  a  solid  rotor  with  the  shaft  projections  bolted  to  the  two  ends, 
(i.e.,  without  a  through  shaft),  is  stiffer  than  a  laminated  rotor 
with  through  shaft.  We  shall  assume  a  solid  rotor  in  this  design, 
although  the  length  of  the  rotor  body  (about  49J^  in.),  being 
less  than  one  and  one-half  times  the  diameter,  would  indicate 
the  feasibility  of  a  rotor  built  up  of  steel  plates. 

Items  (21)  to  (27).  —  On  account  of  our  having  an  odd  number 
of  conductors  per  slot,  we  shall  decide  upon  a  single  layer  winding 
(see  Art.  78,  Chap.  XII).  The  current  density  in  the  armature 
windings  cannot  be  determined  by  the  empirical  formula  (96)  of 
Art.  81,  because  this  is  not  applicable  to  speeds  higher  than  8,000 
ft.  per  minute,  and  in  any  case,  the  conditions  of  cooling  in  an 
enclosed  machine  with  forced  ventilation  are  not  the  same  as 
for  a  self-ventilating  generator.  In  a  turbo-alternator  there  is 
usually  plenty  of  room  for  the  armature  conductors,  the  chief 
trouble  being  with  the  rotor  winding,  which  may  have  to  be 
worked  at  a  high  current  density.  There  is  no  definite  rule  for 
the  most  suitable  current  density  in  the  armature  conductors, 
the  permissible  copper  cross-section  being  dependent  on  the 
length  of  armature  core,  the  position  and  area  of  the  vent  ducts, 
and  the  supply  of  air  that  can  economically  be  passed  through  the 
machine.  The  specific  loading  will  obviously  have  some  effect 
on  the  allowable  current  density  in  the  copper;  and,  as  a  guide 
in  making  a  preliminary  estimate,  we  may  use  the  formula 

160,000 


which  gives  us  for  item  (22)  a  current  density  of  2,000  amp.  per 
square  inch  of  armature  copper. 

It  is  well  to  laminate  the  conductors  in  a  direction  parallel 
to  the  slot  leakage  flux  (see  Art.  88,  page  267,  and  Art.  Ill, 
page  318),  and  we  may  build  up  each  conductor  of  four  flat 
strips  each  %  by  0.14  in.,  giving  a  total  cross-section  of  0.35 
sq.  in.  per  conductor. 

There  will  be  12  copper  strips  in  each  slot,  the  total  thickness, 
including  the  cotton  insulation,  being  about  1.92  in.  The  slot 


EXAMPLE  OF  ALTERNATOR  DESIGN 


329 


insulation  should  be  about  0.16  in.,  or  5^2  m->  thick  (see  Art.  80, 
page  259),  and  the  total  slot  space  for  winding  and  insulation 
will  be  1  in.  wide  by  2^  in.  deep.  The  thickness  of  wedge  might 


A    \B 


re'\ 

*r          ^; 


?'«*• 


A  u-.-! 


,, 


1.64 


1.58 


T 


FIG.  135.— "Developed"  section  through  stator  and  rotor  teeth  of  8000 
k.v.a.  turbo-alternator. 

be  %  in.,  and  we  shall,  in  this  design,  allow  an  extra  slot  depth 
of  1^2  in-  above  the  wedge,  with  a  view  to  increasing  the  slot 
inductance,  and  so  limiting  the  instantaneous  rush  of  current 
in  the  event  of  a  short-circuit.  This  increased  armature  indue- 


330  PRINCIPLES  OF  ELECTRICAL  DESIGN 

tance  might  have  been  obtained  by  using  a  smaller  width  and 
greater  depth  of  copper  conductor;  but,  seeing  that  the  width 
of  tooth  will  probably  be  sufficient,  the  proposed  design  of  slot 
(as  shown  in  Fig.  135)  has  the  advantage  that  the  eddy-current 
loss  in  the  armature  inductors,  from  both  causes  referred  to  in 
Art.  Ill  (page  318),  will  be  very  small. 

The  width  of  copper  strip  was  selected  to  fit  into  the  1-in. 
slot,  because  this  seems  to  provide  a  suitable  cross-section  for 
the  stator  tooth.  Thus,  a  section  halfway  down  the  tooth,  or 
(say)  2  in.  from  the  top,  will  have  a  diameter  of  44  in.,  and  the 

average  width    of  tooth  will  be  v   .g        -  1  =  1.88  in.     On  the 

basis  of  Bg  =  6,000  gausses,  and  a  sinusoidal  flux  distribution 
over  the  pole  pitch,  the  " apparent"  tooth  density  (item  (27)) 

u    i_     vn\la       TT  X  6,000  X  2.62  X  51 

would    be  2BaWn  =  -     2xi.88X46^8-  14'3°°    gaUSS6S 

which  is  not  too  high  (see  Art.  76,  page  251). 

Items  (28)  to  (32). — Assuming  a  flux  density  of  8,500  gausses 
in  the  armature  core  (see  Art.  88,  Chap.  XII),  the  net  radial  depth 
of  stampings  below  the  slots  will  be 

62.2  X  106 

=  11.6  in. 


2  X  8,500  X  6.45  X  48.8 

The  actual  radial  depth  should  be  greater  than  this  to  allow  for 
the  reduction  of  section  due  to  the  presence  of  axial  vent  ducts. 
In  this  particular  machine  it  is  proposed  to  ventilate,  if  possible, 
with  axial  ducts  only,  and  a  fairly  large  cross-section  of  air 
passages  must  therefore  be  allowed.  An  adequate  supply  of  air 
will  probably  be  obtained  if  the  total  cross-section  of  air  duct 
through  the  body  of  the  stampings  (in  square  inches)  is  not  less 
than  0.005  X  cubic  inches  of  iron  in  stator  below  slots.  In  this 
case  the  volume  of  iron  in  the  stator  ring  will  be  approximately 
7r(48.25  +  11.6)  X  11.6  X  46.8  =  102,000  cu.  in.;  and  the  total 
cross-section  of  air  ducts  in  the  stampings  should  be  0.005 
X  102,000  =  510  sq.  in. 

The  actual  radial  depth  of  stamping  below  the  teeth  can  be 
calculated  by  assuming  that  the  air  ducts  reduce  the  gross  depth 

510  f      .     510 

by  an  amount  equal  to  —        : —   —7 —    — '  or  (say)  —     ft0  = 

average  circumference  TT  X  62 

2%  in.  approximately.     Let  us  make  the  depth  Rd  (item  (29))  =  14 
in.,  and  provide  vent  ducts  arranged  generally  as  shown  in  Fig. 


EXAMPLE  OF  ALTERNATOR  DESIGN 


331 


136,  where  there  are  10  holes  per  slot,  each  \Y±  in.  irTdiameter, 
making  a  total  of  ^  (1.25)2  X  10  X  48  =  589  sq.  in. 
The  weight  of  iron  in  core  (item  30)  is 

0.28  X  46.8  X  [7r(38.T252  -  2fl25J)  -  589]  =  28,000  lb.,  approxi- 
mately. 

The  weight  of  the  iron  in  the  teeth  (item  (31))  is 
0.28  X  46.8  X  [7r(24.l252  -  202)  -  (48  X  1  X  4.125)]  =  4,900 lb. 

Taking  the  approximate  flux  densities  as  previously  calculated 
(items  27  and  28),  and  referring  to  the  iron-loss  curve,  Fig.  34, 


Holes  (Total)    U*  Diam.        .- 


FIG.  136. — Armature  stamping  of  8000  k.v.a.  turbo-alternator. 

page  102,  the  iron  loss  per  pound  for  carefully  assembled  high- 
grade  armature  stampings  is  found  to  be  6.1  and  3.2  watts  in 
teeth  and  core  respectively.  The  loss  in  the  teeth  is  therefore 

6.1  X  4,900  =  30,000  watts,  and  in  the  core  below  the  teeth, 

3.2  X  28,000  =  90,000  watts,  making  a  total  of  120  kw.,  or  1.5 
per  cent,  of  the  rated  output,  which  is  not  excessive  although 
quite  high  enough  for  a  machine  of  8,000  kw.  capacity. 

Items  (33)  to  (36). — In  a  machine  of  so  large  an  output  as  the 
one  under  consideration,  the  weight  and  cost  of  copper  should  be 


332  PRINCIPLES  OF  ELECTRICAL  DESIGN 

determined  by  making  a  drawing  of  the  armature  coils  and  care- 
fully measuring  the  length  required.  Since  this  design  is  being 
worked  out  for  the  purpose  of  illustration  only,  we  shall  use  the 
formula  (97)  of  page  261,  and  assume  the  length  per  turn  of 
armature  winding  (item  (33))  to  be 

(2  X  51)  +  (2.5  X  31.42)  +  (2  X  6.6)  +  6  =  199.8  in. 

It  will  be  safer  to  use  the  figure  210  in.  for  this  mean  length ; 
because  all  the  coils  will  probably  be  bent  back  and  secured  in 
position  by  insulated  clamps  in  order  to  resist  the  mechanical 
forces  which  tend  to  displace  or  bend  the  coils  when  a  short- 
circuit  occurs. 

The  cross-section  of  the  conductor  (four  strips  in  parallel)  is 
0.35  sq.  in.,  or  445,000  circular  mils.  The  number  of  turns  per 
phase  is  24,  and  the  resistance  per  phase  at  60°C.  is,  by  formula 

210  V  24 
(21),    page    36,    445^    =  0.01135  ohm.      The  IR  drop  per 

phase  (item  (35))  is  6.01135  X  700  =  7.95,  or  (say)  8.4  volts  in 
order  to  include  the  effect  of  eddy  currents  in  the  conductors. 
The  PR  loss  in  armature  winding  (item  36)  is  3  X  0.01135  X 
(700) 2  =  16,700  watts,  which  should  be  increased  about  25  to  30 
per  cent,  (see  Art.  Ill,  Chap.  XIV)  to  cover  sundry  indeterminate 
load  losses.  The  total  full-load  armature  copper  loss  may  there- 
fore be  estimated  at  21  kw.,.or  0.26  per  cent,  of  the  rated  full- 
load  output;  which  is  about  what  this  loss  usually  amounts  to  in 
a  turbo-generator  of  8,000  k.v.a.  capacity. 

Items  (37)  and  (38). — The  reactive  voltage  drop  per  phase  due 
to  the  cutting  of  the  end  flux  cannot  be  predetermined  accurately; 
but  we  may  use  the  empirical  formula  (99)  page  266,  wherein  the 
symbols  have  the  following  numerical  values. 

The  constant  k  will  be  fairly  high  in  turbo-alternators,  and  we 
shall  assume  the  value  k  =  1.5.  For  the  other  symbols  we  have: 

/  =    60 

p  =  4 
Ts  =  3 
ns  =  4 
Ic  =  700 

In  regard  to  le  and  V ,  the  mean  length  per  turn  (item  (33))  was 
assumed  to  be  210  in.  The  length  le  is  therefore  210  —  2la  or, 
le  =  (210  -  102)  X  2.54  =  275  cm. 


EXAMPLE  OF  ALTERNATOR  DESIGN  333 

The  average  projection  beyond  ends  of  slots  measured  along 
the  side  of  the  coil,  whether  straight  or  bent,  is 


The  reactive  voltage  due  to  cutting  of  end  flux,  with  full-load 
current  per  phase,  is  then 


2.22  X  1.5  X  60  X  4  X  9  X  275  X  (  g)  logio  (12  X  4  X  28.8) 

X  700  X  1C-8  ==  116  volts. 

The  loss  of  pressure  due  to  the  slot  flux  can  be  calculated  as 
explained  in  Art.  97  using  formula  (106)  on  page  288,  wherein 
the  quantities  /,  p,  Tt,  nt,  and  Ic  have  the  same  numerical  values 
as  in  the  formula  for  end-flux  reactive  voltage.  The  remaining 
quantities  are:  the  gross  core  length  la  =  51  X  2.54  ==  129.5  cm., 
d\  =  2%  in.,  and  s  =  1  in.  The  permeances  P2  and  P8  of  the 
flux  paths  in  the  air  spaces  above  the  conductors  can  be  calcu- 
lated as  follows.  Neglecting  the  widening  of  the  slot  to  accom- 
modate the  wedge,  the  permeance  of  the  slot  above  the  winding 

2X1 

(see  Fig.  135)  is  PI  =  — ^ —  =  2,  per  centimeter  length  of  arma- 
ture core  measured  parallel  to  the  shaft.  The  permeance  of  the 
path  from  tooth  top  to  tooth  top  may  be  calculated  by  assum- 
ing the  m.m.f.  of  one  armature  slot  to  set  up  the  flux  in  an  air 
space  of  radial  depth  d  =  %  in.,  and  of  length  X  =  2.62  in. 

7X1 
Thus  PS  =  Q      0  g0  =  0.334.      This  last  quantity  cannot  have 

o  X  ^.D^  ( 

a  numerical  value  smaller  than  as  calculated  by  this  method. 
If  the  rotor  teeth  were  built  up  of  thin  plates  like  the  stator,  the 
numerical  value  of  PS  would  be  greater  than  0.334;  but  we  are 
assuming  a  solid  steel  rotor  (which  is  customary),  and  for  this 
reason  it  will  be  best  to  neglect  the  flux  paths  through  the  iron 
of  the  rotor  teeth.  We  are  concerned  mainly  in  providing  enough 
armature  reactance  to  keep  the  short-circuit  current  within 
reasonable  limits,  and  as  a  sudden  growth  of  leakage  flux  is  im- 
possible in  solid  iron  owing  to  the  demagnetizing  effect  of  the 
eddy  currents  produced,  the  value  for  P3  as  here  calculated  will 
be  about  right. 

By  inserting  all  these  numerical  values  in  formula  (106),  we 
get  for  the  volts  lost  by  slot  leakage  when  the  armature  conduc- 
tors are  carrying  full-load  current,  E.  =  189,  or  (say)  190  volts. 


334  PRINCIPLES  OF  ELECTRICAL  DESIGN 

Items  (39)  and  (40).  —  We  are  now  in  a  position  to  draw  a 
vector  diagram  similar  to  Fig.  115  for  any  power-factor  angle  6, 
the  calculated  numerical  values  of  the  component  vectors  being: 

6,600 


V  3 
EtP  =  8.4 
PEg  =  116 
E0E'Q  =  190 

For  a  given  terminal  voltage  of  6,600  (or  3,810  volts  as  meas- 
ured between  terminal  and  neutral  point)  the  required  developed 
voltage  will  be  a  maximum  when  the  external  power-factor  angle 
is  equal  to  the  angle  EgEtP  because  the  additional  voltage  to  be 
generated  will  then  be  EtEa,  which,  in  this  particular  example, 


c 
FIG.  137. — Vector  diagram  for  8000  k.v.a.  turbo-alternator. 


amounts  to  V(8.4)2  +  (116)2  =  116.4  volts,  the  effect  of  the 
armature  resistance  being  negligible  as  compared  with  the  react- 
ance of  the  end  connections. 

For  80  per  cent,  power  factor,  as  mentioned  in  the  specification, 
the  angle  6  will  be  36°  52',  and  the  developed  voltage  per  phase 
winding  (see  Fig.  137)  is 

OEg  =  \/(OB)*+(BEg)2 
where        OB  =  OA  +  AB  =  OEt  cos  6  +  EtP 
and  BEg  =  BP  +  PEg  =  OEt  sin  0  +  PEg. 

Similarly,  for  calculating  the  full-load  flux  per  pole  (see  Art.  99) 
we  have: 

"Apparent"  developed  voltage  =  OE'g  =  V(OB)2  +  (BE'g)2 


EXAMPLE  OF  ALTERNATOR  DESIGN  335 

Solving  for  the  numerical  values,  we  get: 

OEg  =  3,875  volts  (item  (39)) 
and 

OE'a  ==  4,000  volts. 

The  full-load  flux  per  pole  (item  (40))  is  therefore, 

Item  (17)  X  OE'g  _  62.2  X  106  X  4,000 
OEa  3,810 

=  65.2  X  106  maxwells. 

Item  41. — Since  we  have  decided  upon  a  cylindrical  rotor,  the 
variation  of  flux  density  over  the  pole  pitch  must  be  obtained  by 
distributing  the  field  winding  in  slots  on  the  rotor  surface;  but 
in  the  design  of  salient-pole  machines,  the  pole  face  should  be 
shaped  as  explained  in  Art.  90  (page  269),  and  the  approximate 
dimensions  of  the  pole  core  should  be  decided  upon  with  a  view 
to  providing  sufficient  space  for  the  exciting  coils.  The  cross- 
section  of  the  pole  cores  would  be  determined  as  in  the  case  of 
continuous-current  dynamos  by  calculating  or  assuming  a  leakage 
factor  (see  Art.  103,  Chap.  XIV)  and  deciding  upon  a  flux  density 
in  the  iron  (about  14,000  or  15,500  gausses). 

Items  (42)  to  (44). — Let  us  try  a  rotor  as  shown  in  Fig.  103, 
with  eight  Blots  per  pole,  only  six  of  which  are  wound,  leaving 
two  slots  without  winding  at  the  center  of  each  pole.  The  slot 

vx    OO    OC 

pitch  (item  (43))  is  therefore «~~  "  =  3-76  in.     This  dimen- 

X  X  12 
sion  expressed  in  terms  of  the  stator  diameter  is — g —  =  3.93  in. 

The  slot  width  may  be  decided  upon  by  arranging  for  a  fairly 
high  density  in  the  rotor  teeth.  Thus,  the  open-circuit  flux 
(item  (17))  which  passes  through  a  total  of  eight  teeth  is  62.2  X  106 
maxwells.  If  tr  is  the  average  width  of  rotor  tooth,  in  inches,  the 
average  tooth  density  in  maxwells  per  square  inch  is  B"  = 

8  XL  V  4Q  V  wn^cn  must  be  multiplied  by  „  to  obtain  the  ap- 
proximate maximum  density  in  the  teeth  near  the  center  of  the 
pole.  Neglecting  leakage  flux,  and  assuming  B"max.  =  120,000, 
the  tooth  width  tr  will  be  2.06  in.,  which  indicates  that  a  slot  1% 
in.  wide  will  probably  be  suitable. 

Before  deciding  upon  the  depth  of  rotor  slot,  it  will  be  advisable 
to  calculate  the  equivalent  air  gap  in  order  that  the  field  ampere- 
turns  and  necessary  cross-section  of  copper  may  be  determined. 


336  PRINCIPLES  OF  ELECTRICAL  DESIGN 

The  thickness  of  wedge  for  keeping  the  field  windings  in  posi- 
tion might  be  about  1J^  in.  as  shown  in  Fig.  135;  but  as  the  cen- 
trifugal force  exerted  upon  it  by  the  copper  in  the  slot  may  be 
very  great  on  account  of  the  high  peripheral  velocity,  careful 
calculations  should  be  made  to  determine  the  compression  and 
bending  stresses  in  the  wedge.  The  allowable  working  stress  for 
manganese-bronze  or  phosphor-bronze  wedges  is  about  14,000  Ib. 
per  square  inch. 

Although  we  are  not  designing  a  single-phase  turbo-alternator, 
it  may  be  stated  here  that  a  convenient  means  of  providing 
ammortisseur  or  damping  windings  on  the  rotors  of  single-phase 
machines  (see  Art.  113)  is  to  use  copper  wedges  in  the  slots  and 
connect  them  all  together  at  the  ends  by  means  of  substantial 
copper  end  rings. 

While  discussing  the  matter  of  rotor  slot  design,  the  question 
of  stresses  in  the  rotor  teeth  should  be  mentioned.  After  the 
slot  depth  has  been  decided  upon,  the  centrifugal  pull  on  the 
rotor  tooth  should  be  calculated  and  the  maximum  stress  in  the 
steel  determined,  the  slot  proportions  being  modified  if  this  stress 
exceeds  14,000  Ib.  per  square  inch  for  cast  steel  or  16,000  Ib.  per 
square  inch  for  mild  steel.  The  total  centrifugal  pull  at  the  root 
of  one  tooth  is  due  to  the  weight  of  the  tooth  plus  the  contents 
of  one  slot,  including  the  wedge,  while  the  pull  at  the  narrow 
section  near  the  top  of  tooth  (the  width  W  in  Fig.  135)  is  due  to 
the  contents  of  one  slot  plus  the  wedge  and  the  portion  of  the 
tooth  above  the  section  considered. 

Items  (45)  and  (46) . — The  calculation  of  the  average  permeance 
of  the  air  gap  between  rotor  and  stator  is  carried  out  as  explained 
in  Art.  93  of  Chap.  XIII.  The  approximate  paths  of  the  flux 
lines  are  shown  in  Fig.  135  which  is  a  "  developed"  section  through 
the  stator  and  rotor  teeth;  that  is  to  say,  no  account  is  taken  of 
the  curvature  of  the  air  gap,  the  tooth  pitch  on  the  rotor  being 
made  exactly  equal  to  1.5  times  X,  namely  3.93  in.,  or  10  cm. 
The  actual  air  gap  from  tooth  top  to  tooth  top  (item  (9))  is  d  = 
0.875  in.,  and  if  we  neglect  the  slightly  increased  reluctance  due 
to  the  angle  of  the  tooth  sides  under  the  wedge,  the  component 
flux  paths  may  be  thought  of  as  made  up  of  straight  lines,  or1  of 
straight  lines  terminating  in  quadrants  of  circles.  The  perme- 
ance of  each  section  of  the  flux  path  between  stator  and  rotor 
over  a  space  equal  to  the  rotor  slot  pitch  is  easily  calculated  as  ex- 
plained in  Art.  5,  Chap.  II  (cases  a  and  c).  The  calculated  nu- 


EXAMPLE  OF  ALTERNATOR  DESIGN  337 

merical  values  of  the  permeances  per  centimeter  of  air  gap 
measured  axially  are: 

Path  A  =  0.5725 

Path  B  =  0.408 

Path  B  =  0.408 

Path  C  --=  1.063 

Path  D  =  0.2845 

Path  E  ==  0.323 

Total       =  3.059 

The  permeance  per  square  centimeter  cross-section  of  air  gap 

3  059 
is  therefore  -'         =  0.3059,  and  the  equivalent  air  gap  is  de  = 

0  SOW  =  ^'^  cm'J  or  l'^85  in.  If  great  accuracy  is  required, 
a  similar  set  of  calculations  should  now  be  made  with  the  relative 
position  of  rotor  and  stator  teeth  slightly  changed  so  as  to  bring 
the  center  lines  of  two  teeth  to  coincide,  instead  of  the  center 
lines  of  two  slots  as  shown  in  Fig.  135,  and  the  mean  of  the  two 
calculated  values  will  more  nearly  correspond  with  the  average 
air-gap  permeance.  The  actual  permeance  is  always  somewhat 
greater  than  the  value  obtained  from  calculations  based  on  cer- 
tain conventional  assumptions  regarding  the  flux  paths,  and  we 
may  take  the  equivalent  air  gap  to  be  6e  =  1.25  in. 

Should  any  difficulty  be  experienced  in  calculating  the  per- 
meance of  a  flux  path  such  as  E,  with  curved  flux  lines  at  both 
ends,  it  is  always  permissible  to  divide  it  in  two  parts  as  indicated 
by  the  letters  EI  and  E2  in  Fig.  135,  the  permeance  of  each  part 
being  calculated  separately.  Thus,  in  the  example  which  has 
just  been  worked  out,  the  permeance  of  E\  is  0.686,  and  the  per- 
meance of  E2  is  0.612.  The  total  of  0.323  is  obtained  by  taking 
the  reciprocal  of  the  sum  of  the  reluctances. 

Item  (47).— The  calculations  for  the  curves  of  Fig.  138  have 
been  carried  out  in  the  same  way  as  for  Fig.  81  (item  (70),  Art. 
63,  Chap.  X),  using  formula  (62)  of  Art.  37  for  obtaining  the  ap- 
proximate relation  between  air-gap  and  tooth  densities  when  the 
iron  is  nearly  saturated.  A  curve  must  be  plotted  for  the  rotor 
teeth  as  well  as  for  the  stator  teeth;  indeed  it  is  in  the  rotor  teeth 
that  the  difference  between  "apparent"  and  actual  density  in  the 
iron  will  be  most  marked,  since  it  is  there  that  the  flux  density 
will  attain  the  highest  values.  In  practice  it  will  rarely  be  neces- 


338 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


sary  to  take  account  of  the  effects  of  saturation  in  the  stator 
teeth  of  alternators,  because  with  the  comparatively  low  flux 
densities — especially  in  60-cycle  machines — no  serious  error  will 


ZDOUO 

24000 
28000 
22000 
21000 
20000 
19000 
~  18000 

5   T7fW^ 

/ 

'/ 

/ 

y 

4 

/ 

/ 

J 

/ 

/ 

/  1. 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/  / 

/ 

i 
i 

/ 

/ 

/ 

y 

1 

/ 

/'/ 

'/ 

// 

/ 

Tooth  Density  Bt  (Ga» 

L 

/ 

/ 
/ 

y 

// 

/ 

/ 

// 

/ 

/ 

/ 

$ 

/ 

f 

/ 

/ 

/ 

/ 

/ 

- 

/ 

/ 

/ 

f 

/ 

6000     8000     10000     12000     14000     16000     18000 

Air  Gap  Density  Bg  CGausses) 

FIG.    138. — Tooth    densities   in    terms    of    air-gap    density — 8000   k.v.a. 

turb  o-alternat  or. 

be  introduced  by  using  the  " apparent"  tooth  density  in   the, 
calculations. 

The  depth  of  slot  in  rotor  can  be  approximately  determined  as 
follows : 


EXAMPLE  OF  ALTERNATOR  DESIGN  339 

With  an  equivalent  air  gap  de  =  1.25  in.  and  an  assumed 
sinusoidal  flux  distribution,  the  field  ampere-turns  on  open  cir- 
cuit to  overcome  the  reluctance  of  air  gap  only  will  be 

or.  62.2X10'X,r  1.25X2.54 

"  (51  X  31.42  X  6.45)  X  2  *         0.4ir 

24  000 

which  makes  the  ampere-conductors  per  slot  -—— —  =  8,000. 

o 

This  does  not  include  the.  ampere-turns  to  overcome  the  reluc- 
tance of  the  teeth  and  the  remainder  of  the  magnetic  circuit, 
neither  does  it  take  into  account  the  considerable  increase  of  ex- 
citation with  full-load  current  on  a  power  factor  less  than  unity. 
The  current  density  in  the  copper  may,  however,  be  carried  up 
to  2,500  or  even  3,000  amp.  per  square  inch  of  copper  cross-sec- 
tion, and  it  is  probable  that  a  slot  5  in.  deep  will  provide  sufficient 
space  for  the  field  winding. 

The  numerical  value  of  the  symbol  d  in  formula  (62)  should  be 
something  greater  than  the  actual  clearance  of  %  in.  between 
the  tops  of  the  teeth  on  armature  and  field  magnet,  and  since  the 
difference  between  the  equivalent  and  actual  air  gap  is  %  in., 
we  may  suppose  the  effect  of  slotting  the  surfaces  to  be  equivalent 
to  removing  %$  in.  from  both  stator  and  rotor.  Thus  the  numer- 
ical value  of  6  for  use  in  formula  (62)  will  be  %  in.  -f  % e  m-  = 
iHe  m-  If  we  take  the  tooth  width  at  a  point  halfway  down 
the  tooth,  the  symbols  in  formulas  (62)  and  (63)  have  the  follow- 
ing values: 

d  =*     .0625  in. 

t  =  1.89  in. 

d  =  4. 125  in. 

t  ==  1.64  in. 

d  =  5.0  in. 


Stator 
Rotor 


Since  we  are  considering  the  air-gap  density  over  the  surface  of 
the  stator,  the  slot  pitch  for  the  rotor  has  been  taken  as  X  =  3.93, 
or  one  and  one-half  times  the  stator  slot  pitch.  As  there  are  no 

radial  vent  ducts  in  the  rotor,  the  ratio  T-  in  formula  (62)  may  be 

la 

taken  as 

Axial  length  of  rotor        _  49.5  _ 
Gross  length  of  armature  ~     51 

Item  (48). — With  equally  spaced  slots  around  the  rotor — 
including  the  unwound  portions  of  the  pole  face — only  one  satura- 


340 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


tion  curve  for  air  gap,  teeth  and  slots  need  be  drawn.  This 
curve  (Fig.  139)  is  constructed  as  explained  in  Art.  42,  the  con- 
struction having  previously  been  illustrated  in  connection  with 
the  example  in  dynamo  design  (item  (71),  Art.  63,  Chap.  X). 
The  ampere-turns  per  inch  length  of  tooth  are  read  off  Figs. 
3  and  4;  and  SIMPSON'S  rule  (see  Art.  38,  formula  64)  is  used  in 
calculating  the  ampere-turns  required  for  the  rotor  teeth  at  the 
higher  densities.  In  regard  to  the  stator  teeth,  the  mean  value 
of  the  tooth  density  may  be  used  in  determining  the  ampere- 
turns  required;  the  application  of  SIMPSON'S  rule  being  in  this 


10000 


9000 


^8000 

1 

|  7000 

ft?  6000 
I  5000 


2000 


8000 


10000     12000     14000     16000     18000    20000    22000     24000     26000    28000 

Ampere  Turns  per  Pole 

FIG.  139. — Saturation   curves  for  air-gap   teeth,   and    slots — 8000  k.v.a. 

turbo-alternator. 

case  an  unnecessary  refinement.1  In  calculating  the  tooth 
reluctance  for  plotting  the  saturation  curve  Fig.  138,  no  correc- 
tion for  leakage  flux  has  been  made.  It  is  true  that  the  total 
flux  in  the  body  of  the  rotor  is  somewhat  greater  than  the  useful 
flux  entering  the  armature;  but  the  omission  of  this  correction 
may  be  set  against  the  fact  that  the  tooth  density  calculations 

1  In  most  cases,  the  practical  designer — who  cannot  afford  to  spend  much 
time  on  refinements  of  calculation — calculates  the  density  at  a  section 
one-third  of  the  tooth  length  measured  from  the  narrowest  end,  and  he 
uses  this  value  in  getting  an  approximate  average  value  of  H  from  the  B-H 
curve. 


EXAMPLE  OF  ALTERNATOR  DESIGN  341 

make  no  allowance  for  the  flux  lines  which  pass  from  the  sides 
of  the  tooth  into  the  iron  at  the  bottom  of  the  slot,  thus  causing 
the  actual  density  at  the  narrowest  part  of  the  tooth  to  be  some- 
thing less  than  the  calculated  value. 

The  previously  estimated  depth  of  5  in.  for  the  rotor  slot  seems 
rather  large,  as  it  leaves  hardly  sufficient  section  of  iron  at  the 
root  of  the  tooth.  We  shall  therefore  reduce  this  depth  to  4?4 
in.  as  dimensioned  in  Fig.  135.  The  width  of  the  tooth  at  the 

bottom  is  therefore  -  -  1.625  =  1.2  in. 

oZ 

If  a  larger  section  of  iron  should  be  found  necessary,  it  can 
be  obtained  by  reducing  the  size  of  the  slots  at  the  center  of  the 
pole  face  (i.e.,  those  which  carry  no  field  coils);  but  this  question 
can  be  settled  later. 

The  curve  marked  "air  gap,  teeth,  and  slots"  in  Fig.  139, 
shows  what  excitation  is  required  to  produce  a  particular  density 
in  the  air  gap.  The  departure  from  the  air-gap  line  (the  dotted 
straight  line)  is  due  almost  entirely  to  saturation  of  the  rotor 
teeth,  the  reluctance  of  the  stator  teeth  being  negligible  as  com- 
pared with  that  of  the  l>£-in.  air  path. 

Items  (50)  and  (51). — The  upper  curve  of  Fig.  140  shows  the 
ideal  flux-distribution  curve  for  open-circuit  conditions.  It  is 
a  sine  curve  of  which  the  average  ordinate  is 

62.2  X  106 
*•      6.45  X  31.416  X  51  = 

and  of  which  the  maximum  value  is  therefore  ~  X  6,010  =  9,450 

£i 

gausses.  The  area  of  this  curve  is  a  measure  of  the  total  air- 
gap  flux  on  open  circuit  (item  (17)).  The  pole  pitch — represented 
by  180  electrical  degrees — has  been  divided  into  eight  parts, 
and  the  height  of  the  vertical  lines  is  a  measure  of  the  flux  density 
in  the  air  gap  over  the  center  of  a  rotor  tooth. 

By  providing  a  datum  line  and  vertical  scale  of  ampere-turns 
immediately  below  the  no-load  flux  curve,  it  becomes  a  simple 
matter  to  plot  an  ideal  curve  of  m.m.f.  distribution  over  the 
pole  pitch,  the  shape  of  this  curve  being  such  as  to  produce  the 
desired  flux  distribution  (see  Art.  93,  Chap.  XIII).  It  is  merely 
necessary  to  read  off  the  curve  of  Fig.  139  the  ampere-turns 
corresponding  to  the  required  air-gap  density  and  to  plot  this 
over  the  center  of  the  corresponding  tooth.  In  this  manner 
the  lower  curve  of  Fig.  140  is  obtained.  The  practical  approxi- 


342 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


mation  to  this  ideal  m.m.f.  distribution  would  be  the  arrange- 
ment shown  by  the  stepped  curve,  with  9,000  ampere-conductors 
in  each  slot.  This  would  produce  a  flat-topped  flux-distribution 
curve,  a  condition  which  might  be  remedied  by  putting  5,000 
ampere-turns  in  the  empty  slots  at  the  center  of  the  pole  face; 


"0  22.5U  45U          67.5°    f     90° 

Electrical  Degrees 

FIG.  140. — Ideal  flux-distribution  and  m.m.f.  curves — 8000 
turbo-alternator. 


k.v.a. 


but  such  a  procedure  would  be  very  uneconomical  and  unsatis- 
factory. The  best  thing  to  do  will  be  to  increase  the  permeance 
of  the  center  tooth,  either  by  reducing  the  width  of  the  two  slots 
on  each  side  of  the  center  tooth,  or  by  partly  filling  up  these 
slots  with  iron  wedges  so  shaped  as  to  produce  the  effect  of  a 
tooth  with  parallel  sides.  These  slots  should,  in  any  case,  be 


EXAMPLE  OF  ALTERNATOR  DESIGN 


343 


filled  with  material  equal  in  weight  to  the  copper  and  insulation 
in  the  wound  slots,  in  order  to  improve  the  balance  and  equalize 
the  stresses  at  high  speeds;  and  the  proportion  of  magnetic  to 
non-magnetic  metal  can  be  so  adjusted  as  to  obtain  any  desired 
tooth  reluctance.  If  we  provide  wedges  having  a  thickness  of 
%  m-  a^  the  bottom  of  the  slot,  we  can  get  the  equivalent  of  a 
center  tooth  2.2  in.  wide  with  parallel  sides.  This  calls  for  an 
additional  curve  in  Fig.  139,  which  can  be  calculated  in  the  same 
manner  as  the  curve  previously  drawn,  except  that  the  correction 
for  taper  of  teeth  (SIMPSON'S  rule)  has  not  to  be  applied. 

If  we  decide  upon  a  rotor  winding  with  9,000  ampere-con- 


1ZUUU 

11000 
10000 
9000 
8000 
7000 

f>00fl 

/ 

F 

ux  Curve 

At> 

y 

/ 

^ 

^ 

sX, 

Fl 

nxCun 

ei 

F 

ux 

Cur 

•ve  \A^ 

/ 

/ 

y 

^ 

(8 

0% 

I'UV 

erl 

'act 

or) 

/ 

/ 

// 

\ 

N 

\ 

/ 

/ 

/ 

i 

\ 

^ 

\ 

5000 

/ 

/ 

/ 

^ 

\ 

\ 

\ 

DWU 
4000 

f 

/  / 

/ 

c^ 

s 

\ 

\ 

y, 

/ 

/ 

" 

\ 

\ 

i 

y_ 

r 

8 

\ 

\ 

// 

/ 

CJ 

\ 

\ 

\ 

0 
1000 
2000 
8000 

.    14 

I/ 

/ 

\ 

\ 

/ 

^ 

/ 

\ 

/ 

\ 

0     10    20  80  40    60    60  70    80    90  100  1  10  120  130  140  150  1GO  170  180  10   20    3C 
Electrical  Degrees 
1.  —  Air-gap  flux-distribution  curves  —  8000  k.v.a.  turbo-generato 

FIG 


ductors  per  slot  as  indicated  by  the  stepped  curve  in  the  lower 
part  of  Fig.  140,  we  shall  obtain  an  open-circuit  flux  curve  (A) 
as  plotted  in  Fig.  141.  This  curve  would  be  exactly  similar  in 
shape  to  the  flux  curve  of  Fig.  140  if  it  were  not  for  the  fact  that 
the  widening  of  the  tooth  at  the  center  of  the  rotor  pole  face 
has  lowered  the  reluctance  at  this  point  rather  more  than  would 
have  been  necessary  in  order  to  obtain  the  perfect  sine  curve 
of  flux  distribution.  The  slightly  higher  ordinate  at  the  center 
of  the  new  flux  curve  adds  so  little  to  the  area  of  this  curve  that 
we  shall  not  trouble  to  measure  this.  It  is  evident  that  the 
proposed  excitation  with  9,000  ampere-conductors  per  slot  will 
generate  the  required  open-circuit  voltage. 


344 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


The  m.m.f.  curve  for  flux  curve  A  has  been  re-drawn  in  Fig. 
142,  the  stepped  curve  being  replaced  by  a  smooth  curve.  In 
this  connection  it  should  be  noted  that  the  " fringing"  of  flux  at 
the  tooth  tops  tends  to  round  off  the  sharp  corners  of  the  flux- 
distribution  curves,  and  so  justifies  the  use  of  smooth  curves  in 
any  graphical  method  of  study.  At  the  same  time,  it  will  gen- 
erally be  possible  to  detect  in  oscillograph  records  of  the  e.m.f. 
waves  the  irregularities  or  "ripples"  due  to  the  tufting  of  the 
flux  at  the  tooth  tops;  but  these  minor  effects  will  not  be  con- 
sidered, either  here  or  later  when  calculating  the  form  factor  of 
the  e.m.f.  wave. 


0  10  20  30  40  50  60  70  80  90  100  110  120  130  140  150  160  170  180  10  20  30  40 
.Electrical  Degrees 

FIG.  142.  —  M.m.f.  curves  for  8000  k.v.a.  turbo-generator. 

Items  (52)  to  (54).—  The  area  of  the  flux  curve  A  of  Fig.  141, 
on  the  assumption  that  it  is  a  true  sine  curve  similar  to  the  one  of 
Fig.  140,  and  on  the  basis  of  unit  squares  with  sides  equal  to 
1  cm.,  is  5.91  X  18  =  106.3  sq.  cm.,  where  5.91  is  the  average 
density  in  kilogausses  (item  (18)).  The  required  area  of  the  full- 
load  flux  curve  C,  at  the  specified  power  factor  of  0.8,  is  therefore 


106.3  X 


where  the  figure  4,000  is  the  length  of  the 


vector  OE'g  of  Fig.  137,  as  calculated  under  item  (40),  and  the 
figure  3,810  is  the  open-circuit  star  voltage  (vector  OEt). 

In  order  to  determine  the  field  excitation  necessary  to  provide 
the  required  flux  with  full-load  current  taken  from  the  machine 


EXAMPLE  OF  ALTERNATOR  DESIGN  345 

on  a  power  factor  of  0.8,  it  is  necessary  to  know  the  maximum 
armature  m.m.f.  and  also  the  position  on  the  armature  surface 
(considered  relatively  to  the  field  poles)  at  which  this  maximum 
occurs.  It  was  shown  in  Art.  94,  Chap.  XIII,  that  the  armature 
m.m.f.  can  be  represented  by  a  sine  curve  of  which  the  maximum 
value  (by  formula  (100))  is 

48  X  3  X  700  X  -v/2 
(SI)  a  =  -  v  ,  -  =  11,340  ampere-turns  per  pole. 

7T    X   ^ 

The  displacement  of  this  m.m.f.  curve  relatively  to  the  center  of 
the  pole  is  obtained  approximately  by  calculating  the  angle  0  as 
explained  in  Art.  98.  The  vectors  representing  the  component 
m.m.fs.  have  been  drawn  in  Fig.  137,  the  angle  \f/f  being  calculated 
from  the  previously  ascertained  values  of  the  voltage  vectors 
(see  calculations  under  item  (40)).  Thus 

(0.8  X  3,800)  +  8.4 
COS*  "4,000"  °'763 

whence  V  =  40°  40'. 

Since  27,000  ampere-turns  per  pole  are  required  to  develop 
3,810  volts  per  phase,  and  since  the  saturation  curve  does  not 
depart  appreciably  from  a  straight  line,  the  m.m.f.  vector  OM 
to  develop  OE'g  (i.e.,  4,000  volts)  must  represent  approximately 

27,000X4,000          '  .    . 

—  5~Q1ft       -  =  28,400  ampere-turns,  and  the  required  angle  is, 

o,olU 

tCM. 
'- 


where 

CM0  =  CM  +  MM0  =  OM  sin  f  +  11,340 

=  29,740 
and 

OC  =  OM  cos  ^' 
=  21,650 

The  angle  0  is  thus  found  to  be  53°  57'  or  (say)  54  degrees. 

The  sine  curve  Ma  representing  armature  m.m.f.  can  now  be 
drawn  in  Fig.  142,  with  its  maximum  value  displaced  (54  +  90) 
degrees  beyond  the  center  of  the  pole.  The  required  field 
ampere-turns  are  given  approximately  by  the  length  of  the  vector 
OM0  (Fig.  137),  except  that  the  increased  tooth  saturation  has 

OC 
not   been  taken  into  account.     The  length   OM0  is  -  —  ^  = 


346  PRINCIPLES  OF  ELECTRICAL  DESIGN 

Q  Igg   =  36,800.     An  excitation  slightly  in  excess  of  this  amount 
will  probably  suffice,1  because  if  the  average  density  over  the  pole 

pitch  is  raised  from   6,010  gausses  to   6,010  X  ^?^  =  6,300 

o,olU 

gausses,  the  average  effect  of  increased  tooth  reluctance,  as  shown 
by  Fig.  139,  is  small,  and  we  shall  try  37,000  ampere-turns  on  the 
field.  This  full-load  field  excitation  is  represented  by  the  curve 
Mo  of  Fig.  142.  Now  add  the  ordinates  of  M0  and  Maj  and  ob- 
tain the  resultant  m.m.f .  curve  M.  Using  this  new  m.m.f .  curve, 
we  can  obtain  from  Fig.  139  the  corresponding  values  of  air-gap 
flux  density,  and  plot  in  Fig.  141  the  full-load  flux  curve  C  of 
which  the  area,  as  measured  by  planimeter,  is  found  to  be  112.3 
sq.  cm.  This  checks  closely  with  the  calculated  area  (112  sq. 
cm.)  and  it  follows  that  a  field  excitation  of  37,000  ampere-turns 
will  provide  the  right  amount  of  flux  to  give  the  required  terminal 
voltage  when  the  machine  is  delivering  its  rated  full-load  current 
at  80  per  cent,  power  factor. 

Items  (55)  to  (57). — When  the  shape  of  the  flux  curves  of  Fig. 
141  is  considered  in  connection  with  the  fact  that  a  distributed 
armature  winding  tends  to  smooth  out  irregularities  in  the  result- 
ing e.m.f.  wave  (see  Fig.  118,  page  293),  it  is  evident  that  we  need 
not  expect  any  great  departure  from  the  ideal  sine  curve  in  the 
e.m.f.  waves  of  this  particular  machine  either  on  open  circuit  or 
at  full  load.  At  the  same  time  it  will  be  well  to  illustrate  the 
procedure  explained  in  Arts.  100,  101,  and  102,  by  plotting  the 
actual  e.m.f.  wave  resulting  from  the  full-load  flux  distribution 
curve,  C,  of  Fig.  141. 

The  average  flux  density  corresponding  to  any  given  position 
of  the  four  slots  constituting  one  phase-belt  is  obtained  as  ex- 
plained in  Art.  100,  and  the  instantaneous  values  of  the  "  ap- 
parent" developed  e.m.f.  are  calculated  by  formula  (107).  The 
results  of  these  calculations  have  been  plotted  in  Fig.  143  to 
rectangular  coordinates,  and  in  Fig.  144  to  polar  coordinates. 
The  mean  ordinate  of  Fig.  143  is  3,610  volts,  and  the  r.m.s.,  or 
virtual  value  of  the  e.m.f.,  is  the  square  root  of  the  ratio,  twice 

1  This  method  of  estimating  the  full-load  field  ampere  turns  is  not  scientifically 
sound,  especially  in  the  case  of  salient-pole  machines,  because  the  m.m.f. 
distribution  over  the  armature  surface,  due  to  the  field-pole  excitation,  is 
rarely  sinusoidal  as  here  assumed.  The  correct  increase  of  field  excitation 
to  obtain  a  given  full-load  flux  must,  therefore,  be  obtained  by  trial;  but  the 
method  here  used  indicates  the  approximate  increase  of  excitation  required. 


EXAMPLE  OF  ALTERNATOR  DESIGN 


347 


area  of  the  curve  Fig.  144  -s-  T,  which  is  3,975  volts.     The  form 

Q   in*  C% 

factor  (Art.  101)  is  ~~^  =  1.10.     It  must  not  be  forgotten  that 


6000 

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-K  5000 

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s 

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to  4000 

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s 

\*> 

i 

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uo. 
^*r 

3000 

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, 

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fe  2000 

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— 
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10     20    80    40    60    60     70     80    90    100  110  120  130  140  150  160  170  180    10    20 
Position  of  Center  of  Phase  Belt  on  Pole  Pitch.fElectrical  Degrees) 

Fio.  143. — Wave-shape  of  e.m.f.  developed  in  armature  windings  of  8000 
k.v.a.  turbo-generator  on  80  per  cent,  power  factor  full-load  output. 


CO 


150 


30 


15 


FIG.  144. — E.m.f.  wave  of  Fig.  143  re-plotted  to  polar  coordinates. 

the  machine  under  consideration  is  Y-connected,  and  the  wave- 
shape  of  the   potential   difference  between  terminals  will  be 


348 


PRINCIPLES  OF  ELECTRICAL  DESIGN 


very  closely  represented  by  the  addition  of  two  curves  similar  to 
Fig.  143  with  a  phase  displacement  of  60  degrees,  as  pointed 
out  on  page  291. 

As  a  check  on  the  work,  it  should  be  noted  that  if  the  e.m.f. 
wave  shape  (Fig.  143)  had  been  a  true  sine-wave,  the  virtual 
value  of  the  apparent  developed  voltage  would  have  been  3,605  X 
1.11  =  4,000,  which  proves  the  accuracy  of  the  graphical  work. 
It  is  not,  however,  suggested  that  a  difference  of  1  per  cent,  in 
the  form  factor  is  a  matter  of  practical  importance;  but  in  salient- 


FIG.  145. — Diagrammatic  representation  of  flux  lines  in  turbo-alternator. 


pole  machines  with  incorrectly  shaped  pole  faces  and  a  con- 
centrated armature  winding,  the  wave  shape  may  depart  very 
considerably  from  the  sine  curve,  and  it  is  under  such  condi- 
tions that  the  methods  here  illustrated  will  be  of  the  greatest 
value. 

The  area  of  one  lobe  (Fig.  144),  using  1  volt  as  the  unit  radius 
vector,  is  24,700,000,  and  the  maximum  ordinate  of  the  equiva- 
lent sine-wave,  as  given  by  formula  (108),  is 


,          4X24,700,000  =  5)600yolts 


EXAMPLE  OF  ALTERNATOR  DESIGN  349 

This  is  the  diameter  of  the  equivalent  circle  in  Fig.  144,  and  the 
angle  a,  obtained  as  explained  in  Art.  102,  is  found  to  be  11 
degrees. 

It  is  obvious  that  the  wave  shape  under  no-load  conditions 
will  be  very  nearly  a  true  sine-wave,  and  the  form  factor  will 
therefore  be  approximately  1.11. 

Item  (59). — No  reference  has  been  made  to  item  (58)  because 
this  applies  mainly  to  the  salient-pole  type  of  machine;  the  pro- 
cedure in  proportioning  the  field  poles  and  yoke  ring  being  then 
similar  to  that  followed  in  D.C.  design.  The  depth  of  iron  below 
the  slots  in  the  rotor  of  a  turbo-alternator  is  usually  more  than 
sufficient  to  carry  the  flux,  including  the  leakage  lines.  In  the 
particular  design  under  consideration  we  shall  be  able  to  provide 
air  ducts  at  the  bottom  of  the  rotor  slots  as  shown  in  Figs.  135 
and  145,  and  still  leave  enough  section  of  iron  to  carry  the 
flux. 

The  amount  of  the  leakage  flux  at  the  two  ends  of  the  rotor 
is  not  easily  estimated;  but  when  expressed  as  a  percentage  of 
the  useful  flux  it  is  never  large  in  extra  high-speed  machines  with 
wide  pole  pitch;  the  greater  the  axial  length  of  rotor  in  respect 
to  the  diameter,  the  smaller  will  be  the  percentage  of  the  flux 
leaking  from  pole  to  pole  at  the  ends.  The  rotor  leakage  which 
occurs  from  tooth  to  tooth,  and  in  the  air  gap,  over  the  whole 
length  of  the  machine  is  shown  diagrammatically  in  Fig.  145. 
This  sketch  shows  a  total  of  four  lines  of  leakage  flux  which  pass 
through  the  body  of  the  rotor,  but  do  not  enter  the  armature. 
This  leakage  flux  will  not  appreciably  affect  the  flux  density  in 
the  rotor  teeth  near  the  neutral  zone,  because  it  will  follow  the 
path  of  least  reluctance  and  be  distributed  between  several 
teeth. 

The  calculation  of  the  rotor  slot  flux  may  be  carried  out  as  for 
the  stator  windings.  Thus,  at  full  load,  with  12,300  ampere- 
conductors per  slot,  the  flux  passing  from  tooth  to  tooth  below 
the  wedge  is 

0.47T  X  12,300       (3.5  X  49.5)  X  2.54 

— ~  -  =  2,100,000  maxwells. 


Average  m.m.f.  Permeance 

The  flux  in  the  space  occupied  by  the  wedge  and  insulation 
above  the  copper,  including  an  allowance  for  the  spreading  of  the 


350  PRINCIPLES  OF  ELECTRICAL  DESIGN 

flux  lines  into  the  air  gap  above  the  wedge  (tooth  top  leakage)  ,  is 
approximately, 

0.47T  X  12,300  X  (L75  X  *9'5)  X  2<54  =  1,900,000  maxwells, 

l.o 

where  the  figure  1.75  is  the  assumed  radial  depth,  in  inches,  of 
the  flux  path,  and  1.8  is  the  assumed  average  length  of  the  flux 
lines  (somewhat  greater  than  the  width  of  slot  below  the  wedge). 
The  sum  of  the  two  flux  components  is  4,000,000  maxwells, 
making  the  total  slot  flux  for  both  sides  of  the  pole  face  equal  to 
twice  this  amount,  or  8,000,000  maxwells.  As  a  rough  estimate, 
we  may  assume  the  end  leakage  to  be  about  one-sixth  of  this, 
making  a  total  of  9,300,000  maxwells.  The  full-load  leakage 

65  21  93 

coefficient  is  therefore  -  '      0   '-  =  1.142  or  (say)  1.15. 

oo.J 

The  maximum  number  of  flux  lines  in  the  rotor,  which  cross 
the  section  below  the  slots  (represented  by  28  lines  in  Fig.  145) 

x  LIB  -  87,600,000 


The  cross-section  of  iron  below  the  vent  ducts  is  12%  X  49^  = 
600  sq.  in.;  which  makes  the  average  flux  density  62,500  lines 
per  square  inch.  The  section  of  iron  below  the  slots  is  therefore 
sufficient,  and  the  reluctance  of  the  body  of  the  rotor  is  a  neg- 
ligible quantity  in  comparison  with  that  of  the  teeth  and  air  gap. 

Items  (62)  to  (68).  —  With  a  density  of  8,500  gausses  in  the 
stator  core  (item  28),  and  ample  iron  section  in  the  rotor,  the 
additional  ampere-turns  required  to  overcome  reluctance  of 
armature  and  field  cores  will  probably  not  exceed  200;  and  since 
this  is  a  very  small  percentage  of  the  excitation  for  air  gap  and 
teeth,  we  shall  not  need  to  draw  a  new  curve  for  item  (62): 
the  curves  of  Fig.  139  may  be  thought  of  as  applying  to  the 
machine  as  a  whole.  The  ampere-turns  at  no  load  and  at  full 
load  (items  63  and  64)  will  therefore  be  taken  at  27,000  and 
37,000  respectively,  as  previously  calculated. 

The  slot  insulation  should  be  about  %  in.  thick,  and  the  field 
winding  might  be  in  the  form  of  copper  strip  1J4  in.  wide  laid 
flat  in  the  slot.  Allowing  %  in.  total  depth  of  insulation  —  prefer- 
ably of  mica  or  asbestos  fabric  —  between  the  layers  of  the  wind- 
ing, the  cross-section  of  copper  will  be  2%  X  1M  =  3.6  in. 

12  300 
making  the  current   density  at  full  load,  A  =  =  3,430 


EXAMPLE  OF  ALTERNATOR  DESIGN  351 

amp.  per  square  inch.     This  is  a  high,  but  not  necessarily  an 
impossible  figure. 

The  mean  length  per  turn  of  the  rotor  winding  should  be  meas- 
ured off  a  drawing  showing  the  method  of  bending  and  securing  the 
end  connections.  We  shall  assume  this  length  to  be  156  in.  All 
the  turns  will  be  in  series,  and  the  mean  length  per  turn  for  the 
four  poles  in  series  will  be  156  X  4  =  624  in.  Assuming  the 
potential  difference  at  the  slip  rings  to  be  120  volts,  the  cross- 
section  of  the  winding,  by  formula  (26),  is 


(m)  =  -  ==  192,500  or  0.1512  sq.  in. 


If  we  use  a  copper  strip  0.12  in.  thick,  the  number  of  conductors 

2  875 
in  each  slot  will  be  -/rro"  :=  24,  making  the  turns  per  pole  24  X  3 

=  72. 

37  000 
The  current  per  conductor  at  full  load  must  be  ~n^~  =514 

514 

amp.,   whence  the  current  density  is  A  =  roK~\      To  =  3,430 

i.zo  /s  u.i  & 

amp.  per  square  inch. 

72  X  4  X  156 
The  total  length  of  copper  strip  is  -     —  ;-„  —     -  =  3,740  ft. 

The  resistance  (hot)  will  be  about  0.250  ohm,  and  the  required 
pressure  at  slip  rings  will  be  0.25  X  514  =  128.5  volts.  The 
PR  loss  is  therefore  128.5  X  514  =  66  kw.,  or  0.825  per  cent. 
of  the  rated  output.  This  is  rather  on  the  high  side  for  so  large 
a  machine,  and  it  may  be  accounted  for  by  the  fact  that  the  air 
gap  is  perhaps  somewhat  greater  than  it  need  be;  but  the  ef- 
ficiency will  not  be  affected  appreciably. 

Item  (69).  —  The  cooling  air,  which  enters  at  one  end  of  the 
machine,  is  supposed  to  travel  through  the  longitudinal  vent 
ducts  to  the  other  end  of  the  machine,  no  radial  ducts  being 
provided.  Such  an  arrangement  leads  to  the  temperature  of 
one  end  of  the  machine  being  higher  than  the  other  end;  but 
systems  of  ventilation  designed  to  obviate  this  are  usually  less 
simple,  and  the  straight-through  arrangement  of  ducts  has  much 
to  recommend  it.  In  machines  larger  than  the  one  under  con- 
sideration, it  might  be  necessary  to  have  the  cold  air  enter  at 
both  ends,  in  which  case  one  or  more  radial  outlets  would  be 
provided  at  the  center. 

In  addition  to  the  ducts  of  which  mention  has  already  been 


352  PRINCIPLES  OF  ELECTRICAL  DESIGN 

made,  we  may  provide  a  number  of  spaces  between  the  stator 
iron  and  the  casing  to  allow  of  air  being  passed  over  the  outside  of 
the  armature  core.  Let  us  suppose  that  there  are  twelve  such 
ducts,  each  10  in.  wide  by  1  in.  deep;  the  total  cross-section  of 
the  air  ducts  is  then  made  up  as  follows: 

Outside  stator  stampings 12  X  10  X  1  =  120  sq.  in. 

Holes  punched  in  stator  stampings  (Fig.  136) ==  590 

Spaces  above  wedge  in  stator  slots  (Fig.  135) 

48  X  1.5  X  1  =    72 

Clearance  between  stator  and  rotor   %  X  TT  X  39^  =108 
Spaces  in  rotor  forging  below  slots 32  X  1.94  =    62 

Total =  952  sq.  in. 

or  6.6  sq.  ft. 

The  total  losses  in  the  machine,  without  including  windage  and 
sundry  small  losses,  are 

Total  core  loss  (item  (32)) 120  kw. 

Stator  PR  loss  (item  (36)) 21  kw. 

Rotor  PR  loss 66  kw. 

Total 207  kw. 

If  we  allow  100  cu.  ft.  of  air  per  minute  for  each  kilowatt  dissi- 
pated (see  Art.  33,  Chap.  VI),  it  will  be  necessary  to  pass  20,700 
cu.  ft.  of  air  through  the  machine  per  minute.  This  makes  the 

20  700 
velocity  in  the  vent  ducts      '  „      =  3,140  ft.  per  minute,  which 

is  well  below  the  permissible  limit. 

Item  (70). — With  varying  degrees  of  tooth  saturation — espe- 
cially when,  as  in  this  design,  all  the  teeth  are  not  of  the  same 
cross-section — the  only  correct  method  of  predetermining  the 
open-circuit  saturation  curve  (similar  to  Fig.  124,  Art.  104),  is  to 
plot  the  flux  distribution  for  different  values  of  the  exciting 
ampere-turns,  and  calculate  the  e.m.f.  developed  in  each  case. 
It  is  not  necessary  to  calculate  a  large  number  of  values  in  this 
manner;  two  or  three  points  taken  with  fairly  high  values  of  the 
exciting  current  will  show  how  the  tooth  saturation  affects  the 
resulting  flux;  and  a  curve  can  be  drawn  connecting  the  known 
straight  part  of  the  saturation  curve  with  these  ascertained  values 
for  the  higher  densities. 

The  saturation  curve  for  zero  power  factor  can  be  drawn  as 
explained  in  Art.  106  (Fig.  125),  and  the  construction  of  Figs. 


EXAMPLE  OF  ALTERNATOR  DESIGN  353 

129  and  130  can  be  applied  for  obtaining  curves  giving  the  ap- 
proximate connection  between  terminal  volts  and  exciting  current 
for  any  other  power  factor.  We  shall  confine  ourselves  here  to 
calculating  the  inherent  regulation  by  the  more  correct  method  as 
outlined  in  Art.  110,  and  since  much  of  the  work  has  already 
been  done  in  connection  with  full-load  current  on  80  per  cent, 
power  factor,  this  is  the  condition  which  we  shall  choose  for  the 
purpose  of  illustration. 

We  know  that  although  27,000  ampere-turns  per  pole  will 
develop  the  specified  terminal  voltage  when  no  current  is  taken 
from  the  machine,  this  excitation  must  be  increased  to  37,000 
ampere-turns  to  give  the  same  terminal  voltage  under  full-load 
conditions  (80  per  cent,  power  factor).  If  then,  we  can  calculate 
the  voltage,  with  this  greater  field  excitation,  when  the  load  is 
thrown  off,  the  inherent  regulation  can  be  predetermined,  and, 
incidentally,  we  shall  obtain  a  point  on  the  open-circuit  charac- 
teristic corresponding  to  a  fairly  high  value  of  the  excitation. 

The  required  flux  curve,  marked  A0,  has  been  plotted  in  Fig. 
141.  It  is  derived,  like  any  other  flux  curve,  from  the  m.m.f. 
curve  M0  of  Fig.  142,  by  using  the  saturation  curves  of  Fig.  139 — 
which  must  be  extended  beyond  the  limits  of  the  diagram  in 
order  to  read  the  flux  values  for  the  higher  degrees  of  excitation. 
Careful  measurements  of  the  flux  curve  A0  give  an  area  over  the 
pole  pitch  of  129  sq.  cm.,  and  if  we  assume  the  form  factor  of 
the  resulting  e.m.f.  wave  (not  plotted)  to  be  the  same  as  for  the 
open-circuit  curve  at  normal  voltage,  i.e.,  1.11,  the  voltage  cor- 
responding to  the  flux  Ao  will  be 

3,810  X  119 
106.3 

where  106.3  is  the  previously  measured  area  of  flux  curve  A. 
The  inherent  regulation  at  80  per  cent,  power  factor  is  therefore 

4,625  -  3,810 

3,810  =21.4  per  cent. 

This  is  well  within  the  specified  limit  of  25  per  cent,  which 
again  points  to  the  fact  that  a  somewhat  smaller  air  gap,  or  a 
lower  flux  density  in  the  rotor  teeth  would  have  been  permissible. 

It  should  be  pointed  out  here  that  the  external  power  factor 
corresponding  to  the  flux  distribution  curve  C  of  Fig.  141  is  not 
necessarily  exactly  0.8;  because  the  method  of  determining  the 

23 


354  PRINCIPLES  OF  ELECTRICAL  DESIGN 

angle  /3  =  54  degrees  (Figs.  137  and  142)  was  based  on  certain 
conditions  that  may  not  actually  be  fulfilled.  A  closer  determina- 
tion of  the  external  power  factor  corresponding  to  the  conditions 
that  have  been  studied  is  easily  made  as  follows. 

The  angle  a  of  Fig.  137  was  determined  on  page  349  and  found 
to  be  11  degrees.  The  angle  \f/f  is  therefore  0  —  a  =  54  —  11  = 
43  degrees,  instead  of  the  previously  obtained  value  of  40°  40' 
(see  calculations  Under  items  (52)  to  (54)).  The  vector  OE'g  is 
known,  and  its  value  is  4,000  volts,  since  this  is  the  e.m.f.  devel- 
oped by  the  flux  distribution  C  of  Fig.  141.  In  order  to  obtain 
the  corrected  values  for  the  angle  6  and  the  length  of  the  vector 
OEt  in  Fig.  137,  we  have: 

OB  =  OE'g  cos  ^'  =  4,000  cos  43° 

=  2,925 

whence  OA  =  2,917 

Also,  BE'g  =  OE'g  sin  V  =  4,000  sin  43° 

=  2,725 
whence          AEt  =  2,419 

Thus,  tan  6  =  ~r=  =  0.83,  which  corresponds  to  0  =  39°  40'. 


The  external  power  factor  is  therefore  cos  0  =  0.77  and  the 

terminal  voltage  =  \/3  ^-^  =  --  (JTT~  "  =  6)57°* 

Thus,  the  condition  that  has  actually  been  worked  out  by 
graphical  methods  corresponds  to  a  terminal  voltage  of  6,570  with 
full-load  current  on  an  external  power  factor  of  0.77.  This  is 
sufficiently  close  to  the  specified  condition  with  figures  6,600  and 
0.8  to  show  that  the  machine  will  comply  with  the  requirements. 
Applying  these  corrections  to  the  regulation,  we  have: 


Inherent  regulation  with  full-load  current 
on  an  external  power  factor  of  0.77 


_  A/3  X  4625  -  6570 

~  6570 
=  22  per  cent. 

Item  (71). — On  the  assumption  that  the  inductance  of  the 
armature  windings  can  be  correctly  calculated,  the  short-circuit 
current  corresponding  to  any  given  field  excitation  can  readily  be 
determined  by  the  method  described  in  Art.  107.  The-  curve 
marked  volts  in  Fig.  146  is  the  open-circuit  characteristic  of  the 
machine,  the  scale  of  ordinates  being  on  the  left-hand  side  of  the 
diagram.  The  construction  shows  that,  in  order  to  develop  the 


EXAMPLE  OF  ALTERNATOR  DESIGN 


355 


flux  necessary  to  produce  full-load  current  (700  amp.)  in  the 
short-circuited  windings,  1,900  ampere-turns  per  pole  are  re- 
quired. This  is  the  excitation  which  will  develop  an  "  apparent " 
e.m.f.  equal  to  the  sum  of  items  (37)  and  (38);  the  effect  of  the 
IR  drop  being  negligible.  The  armature  m.m.f.  is  almost  directly 
demagnetizing,  and  the  ampere-turns  per  pole  must,  therefore,  be 
1,900  +  11,340  ==  13,240.  The  current  curve,  within  the  range 
of  the  diagram,  will  be  a  straight  line,  and  with  the  full-load  ex- 
citation of  37,000  ampere-turns,  the  short-circuit  current  will  be 
1,950  amp.,  or  2.8  times  normal.  This  is  the  steady  value  which 
the  short-circuit  current  would  attain  if  the  field  excitation  were 
gradually  brought  up  to  full-load  value;  but  at  the  instant  of  the 
occurrence  of  a  short-circuit  with  full-load  excitation,  the  current 


Volts  per  Phase  Winding 

2000  2 
1800  g 
1600  ^ 
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1000  « 
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^ 

s" 

v\ 

^C-, 

•  

— 

_L 

_L 

0       4000     8000     12000  16000   20000  24000   28000  82000  36000   40000 
Ampere  Turns  per  Pole 

Fio.  146.  —  Curves  of  open-circuit  voltage  and  short-circuit  current  —  8000 
k.v.a.  turbo-generator. 

would  be  limited  only  by  the  impedance  of  the  stator  windings 
which  must  set  up  a  flux  of  self-induction  equal  to  the  total  flux. 
If  we  neglect  the  effect  of  iron  saturation  and  the  changes  in  the 
paths  of  the  flux  leakage  lines,  the  momentary  current  might  be 

4  000 
700  X  ~OC"  =  9,150  amp.,   or   13  times  normal-load  current. 


It  might  even  be  greater  than  this,  depending  upon  the  instan- 
taneous value  of  the  generated  e.m.f.  when  the  short-circuit  oc- 
curs, and  the  windings  should  be  arranged  if  possible  to  with- 
stand without  injury  the  mechanical  forces  exerted  on  the  coils 
under  this  condition.  If  this  cannot  be  done,  reactance  coils 
external  to  the  machine  must  be  provided;  but  the  tendency 
to-day  is  to  design  machines,  even  of  the  largest  sizes,  with  suffi- 


356  PRINCIPLES  OF  ELECTRICAL  DESIGN 

cient  internal  magnetic  leakage  to  prevent  mechanical  injury 
due  to  excessive  magnetic  forces  on  short-circuit. 

Items  (72)  and  (73). — Seeing  that  the  calculation  of  efficiency 
under  different  conditions  of  loading  was  illustrated  in  Art.  63 
when  working  out  the  example  in  dynamo  design,  it  will  not  be 
necessary  to  cover  the  same  ground  a  second  time.  We  shall, 
therefore,  determine  the  full-load  efficiency  only  (item  (73)). 

The  windage  and  bearing  friction  loss  is  very  difficult  to  esti- 
mate; but  some  approximate  figures  were  given  in  Art.  111.  We 
shall  assume  1.1  per  cent,  of  the  total  k.v.a.  output  for  the  loss 
due  (mainly)  to  the  friction  of  the  air  passing  through  the  axial 
ducts,  and  also  to  friction  in  the  outside  bearing.  It  is  assumed 
that  the  losses  in  the  bearing  on  the  side  of  the  prime  mover  are 
included  in  the  steam-turbine  efficiency.  The  power  necessary 
to  drive  the  blower  is  not  included  in  the  above  estimate  of  the 
windage  and  friction  loss. 

For  the  correct  calculation  of  iron  losses,  the  reader  is  referred 
to  Art.  60,  Chap.  IX  where  the  method  of  determining  the  tooth 
losses  was  explained ;  but  since  the  maximum  tooth  density  under 
full-load  conditions,  as  indicated  by  curve  C  of  Fig.  141  does  not 
differ  appreciably  from  the  maximum  of  the  open-circuit  curve 
(A),  we  shall  not  trouble  to  correct  the  tooth  losses  as  previously 
calculated. 

With  reference  to  the  iron  in  the  body  of  the  stator,  the  flux 
indicated  by  the  area  of  the  load  flux  curve  C  of  Fig.  141  does 
not  all  enter  the  core  below  the  slots,  because  this  total  flux  in- 
cludes the  slot-leakage  flux,  as  explained  in  Art.  95.  The  posi- 
tion of  the  conductors  carrying  the  maximum  current  coincides 
with  the  zero  point  on  the  armature  m.m.f.  curve.  This  is  the 
position  17  degrees  in  Figs.  141  and  142;  and  the  current  in  the  con- 
ductor will  be  approximately  700  X  cos  53°  =  420  amp.  The  slot- 
leakage  flux  corresponding  to  this  particular  current — the  total 
not  the  ''equivalent"  flux — can  be  calculated  as  explained  in 
Art.  96,  Chap.  XIII,  when  deriving  formula  (104)  which,  how- 
ever, gives  the  " equivalent"  and  not  the  total  slot  leakage  flux. 
If  $>s  is  the  calculated  slot  flux,  and  $  is  the  total  flux  per  pole 
in  the  air  gap,  then  the  flux  actually  carried  by  the  section  of 

<i> 
the  armature  iron  below  the  slots  is  ^  —  $«•     This  correction 

is  a  refinement  which  need  not  be  applied  in  the  case  of  a  turbo- 
alternator,  in  which  the  pole  pitch  is  always  large,  causing  <£, 


EXAMPLE  OF  ALTERNATOR  DESIGN  357 

to  be  small  in  relation  to  $;  but  in  machines  with  a  small  pole 
pitch  —  especially  if  there  is  only  one  slot  per  pole  per  phase— 
the  correction  should  be  made. 

The  full-load  flux  per  pole  is  65.2  X  106  maxwells  (item  (40)), 
as  against  62.2  X  108  (item  (17))  on  open  circuit.  With  the  in- 
crease of  flux  density,  the  loss  per  pound  of  iron  will  be  about 
3.5  instead  of  3.2  watts,  and  the  full-load  iron  loss  will,  therefore, 
be  0.3  X  28,000  =  8.4  kw.  more  than  on  open  circuit;  thus 
bringing  the  total  iron  loss  up  to  (say)  130  kw. 

The  loss  at  the  slip  rings  may  be  calculated  by  the  approximate 
formula  given  in  Art.  111.  The  diameter  of  the  slip  rings  will 
probably  not  be  less  than  15  in.,  so  that  the  rubbing  velocity 

will  be  —,  -^  X  1,800  =  7,100  ft.  per    minute.     The  contact 
l« 

area  of  the  two  sets  of  brushes  (to  carry  514  amp.)  might  be  5 

7  100  X  5 
sq.  in.,  making  the  loss  from  this  cause  "'"I""  =  355  watts, 


which  is  negligible  in  comparison  with  the  other  losses. 
Adding  up  the  separate  losses,  we  have: 

Windage  and  friction  .................................  88  kw. 

Stator  iron  .........                                                                 .  130  kw. 

Stator  copper  .........  21  kw. 

Rotor  copper  ....................                              ......  66  kw. 


Total 305  kw. 

The  kw.  output  is  0.8  X  8,000  =  6,400  and  the  efficiency,  ex- 
cluding losses  in  exciter  and  in  air  blower  external  to  the  gener- 

ator,  is  therefore  ^^SOS  =  °'955- 

Items  (74)  and  (75). — On  the  basis  of  290  kw.  to  be  carried 
away  by  the  circulating  air  with  a  mean  increase  of  temperature 
of  about  20°C.,  the  quantity  of  air  required  will  be  29,000  cu.  ft. 
per  minute.  The  cross-section  of  the  ducts  (item  (69))  is  6.6  sq. 
ft.,  and  the  average  velocity  in  the  ducts  (item  (75))  is,  therefore, 

'        =  4,400  ft!  per  minute. 

In  the  design  of  large  generators  there  are  many  matters  of 
detail  to  be  considered  which  have  received  but  little  attention 
here.  The  question  of  temperature  rise,  for  instance,  is  one  that 
would  receive  more  attention  from  the  practical  designer  than 
we  have  given  it  here.  It  is  usually  permissible  to  assume  that, 
if  the  difference  in  temperature  between  ingoing  and  outgoing  air 


358  PRINCIPLES  OF  ELECTRICAL  DESIGN 

is  20°C.,  the  actual  temperature  rise  of  the  heated  surfaces  as 
measured  by  thermometer,  will  not  exceed  40°  to  50°C.;  but, 
unless  every  part  of  the  machine  is  carefully  designed,  excessive 
local  heating  may  result.  The  temperature  of  the  copper  in  the 
slots  might  ordinarily  be  from  15°  to  25°C.  higher  than  that  of 
the  iron  from  which  it  is  separated  by  layers  of  insulating  material ; 
but  should  this  insulation  be  very  thick,  and  the  cooling  ducts 
of  insufficient  section  or  improperly  located,  very  much  higher 
internal  temperatures  may  be  reached. 


INDEX 


Acyclic  D.  C.  generators,  reference 

to,  83 

Air-gap,  density  (see  Flux  density), 
"equivalent,"  117,  212,  337 
length  of,  119,  252,  325,  337 
permeance,  116,  272,  336 
Alternator,  design  sheets  for  8,000 

k.v.a.,  322 

three  phase,  output  of,  247 
Alternators,  classification  of,  238 

single-phase,  280,  321 
Amortisseur  windings  (see  Pole-face 

windings). 
Ampere-conductors    per    inch     (see 

Specific  loading). 
Ampere-turns,    calculation    of,    on 

horseshoe  magnet,  60 
in  dynamo  field  coils,  139,  187 
in  series  winding  (dynamo),  190 
on  armature  (of  dynamo),  78,  136 

(of  alternator),  249,  278,  345 
on  interpoles,  calculation  of,  174 
per  unit  length  in  air,  21 
"Apparent"    developed    voltage 

(A.C.),  287,  334 

Armature  ampere-turns  (see  Ampere- 
turns). 

coils,  length  of,  97,  261 
conductors,  current  density  in,  97, 

259 

insulation  of  (see  Insulation), 
losses  in,  318 
number  of,  206 
core,  flux  densities  in,    104,  210, 

266,  330 

losses  in,  100,  104,  266,  331,  356 
net  and  gross  length  of,  103,  326 
current,  effect  of,  on  flux   distri- 
bution, 135 

diameter    (how   determined),   77, 
205 


Armature,  m.m.f.,  136,  274,  278 

in  single-phase  alternators,  280 
reactance  and  reaction,  304 
teeth  (see  Teeth), 
windings,  concentrated  (.A.C.),  242 
distributed  (A.C.),  242 
double  layer  (A.C.),  255 
duplex,  86 
full  pitch,  85 
inductance  of,  263 
insulation  of  (see  Insulation), 
losses  in,  99 
multiple  or  lap,  86 
multiplex,  86 
resistance  of,  97,  209,  260 
series  or  wave,  86 
short  pitch  (chorded),  85,  160 
simplex,  86 

single  layer  (A.C.),  255 
single-,  two-,  and  three-phase, 

240 
spread  of  (A.C.),  257;  in  single 

phase,  258 

Armatures,  drum-wound,  84 
ring-wound,  84 

temperature  rise  of,  107,  111,  267 
ventilation  of,  105,  261,  351 
AKNOLD,  DR.  E.,  referred  to,  141 
Asynchronous   A.C.   generators   re- 
ferred to,  237 

B 

Balancing  coils  (see  Pole-face  wind- 
ings). 

BEHREND,  B.  A.,  referred  to,  307 
B-H  curves,  16,  17,  18,  215 
Brown  and  Sharp  gage,  34,  36 
Brush  contact  resistance,  179,  226 
effect    of,    on    commutation, 

147,  175 

pressure,  usual,  179 
width,  as  affecting  commutation, 
157,  162 


359 


360 


INDEX 


Brush,  width,  usual  limits,  178 
Brushes,  current  density  at  contact 
surface  of,  178,  180 

friction  losses  at,  182,  227,  318 

PR  losses  at,  181 


Calorie,  definition  of,  47 
CARTER,  F.  W.,  referred  to,  116 
Chorded  armature  windings,  85,  160 
Circular  lifting  magnet,  design  of,  64 
mils,     definition    35,     conversion 

factor,  36 
per  ampere,  conversion  factor, 

41 

CLAYTON,  A.  E.,  referred  to,   309 
Clutch,  magnetic,  50 
Coefficient     (see    Cooling,  Friction, 

Leakage,  coefficient). 
Commutating  poles  (see  Interpoles). 
Commutation,    effect   of   brush   re- 
sistance on,  147,  175 
of  end  flux  on,  151 
of  field  distortion  on,  169 
of  slot  leakage  on,  149 
ideal,  or  "straight  line,"  148 
mechanical  details  affecting,  178 
selective,  88 
sparking  limits,  175 
theory  of,  142 
time  of,  140,  162 
Commutator,  180,  184 
diameter  of,  181 
peripheral  velocity  of,  184 
segments,  number  of,  90,  93,  223 

volts  between,  94 
temperature  rise  of,  181 
Compensating   windings    (see   Pole- 
face  windings). 

Concentrated  windings    (see  Arma- 
ture windings). 

Conductors   (see  Armature  conduc- 
tors). 

Conical  pole  faces,  magnets  with,  31 
Cooling  coefficient  (magnet  coils),  45 

(dynamo  field  coils),  194 
of  armatures,  etc.  (see  Tempera- 
ture rise). 


Copper  wire,  properties  of,  33 
resistance  of,  36 
tables,  34,  35,  38 
weight  of,  36,  47 
Core  (see  Armature  core). 
Cost,  important  factor  in  design,  6 

of  electromagnets,  49,  69 
Critical  speed  of  rotor,  327 
Cultural    subjects,    value    of  study 

of,  8 
Current  density   at  brush    contact 

surface,  178,  180 
in    armature    conductors,     96, 

259,  328 

in  coils  of  horseshoe  magnet,  56 
usual,  in  magnet  windings,  41 


Damping  grids  (see  Pole-face  wind- 
ings). 

"Dead"    coils    in    armature    wind- 
ings, 90 

Density  (see  Flux,  Current,  density). 

Distributed  armature  windings  (see 

Armature), 
field  windings,  270,  335 

Distribution  factor,  243 

Diverter,    193,    233,   for  interpoles, 
168 

Dynamo,     design     sheets     for     75 
kw.,   201 

A-connection,  244 


E 


Eddy-current  losses  (see  Losses). 
Efficiency  of  dynamo,  195,  197,  233 

of  alternator,  317,  357 
Electromagnets,  circular  type,  64 
cost  of,  49,  69 
design  of,  48,  53 
short  stroke  plunger  type,  64 
E.m.f.  (see  also  Voltage). 

developed  in  alternator  armatures, 
•       242,  261,  291 
in  dynamo  armatures,  72,  138 
by  cutting  of  end  flux  (commu- 
tation), 159 


INDEX 


361 


E.m.f.,  instantaneous    value  of  de- 
veloped (alternator),  293 
virtual,  or  effective,  value  of,  294, 

346 
wave  form,  291,  346 

Empirical   formulas,    impossible    to 
avoid,  3 

Enamelled  wires,  38 

End  connections  (armature),  length 
of,  98 

End  flux  cut  by  commutated  coil, 
155 

English    language,     importance    of 
thorough  knowledge  of,  5 

Equalizing    connections    for    dyna- 
mos, 90 

Equivalent  sine  waves,  295,  348 
slot  flux,  161,  164,  285 

ESTERLINE,    PROP.   J.    W.f  referred 
to,  101 

Excitation  (see  Ampere-turns). 

Exciting   current,    usual   values   of 
(dynamos),  192 


Face  conductors  (definition),  72  (see 

also  Armature  conductors). 
Factor  of  safety,  calculation  of,  in 

magnet  design,  63 

Fans,  power  required  to  drive  venti- 
lating, 107,  318 

FIELD,  PROP.  A.  B.,  referred  to,  319 
Field  distortion  in  relation  to  com- 
mutation, 169 

magnet  design,  185,  228,  299 
with  distributed  windings,  270 

272,  335 
rheostat,  191 
windings,  design  of,  191,  231,  299, 

351 
FLEMING  AND  JOHNSON,  referred  to, 

192 
Flux  density  (definition),  13 

in  air  gap  of  alternators,  251, 

325 

of  dynamos,  75,  132 
of  magnets,  50,  54,  63,  65 
in  armature  cores.  104,  210,  266, 
330 


Flux  density  in  armature  teeth,  102, 
104,  119,  122,  197,  209,  214,  251, 

266 
in  commutating  zone,  165,  167, 

224 

in  pole  core,  213 
in  rotor  teeth,  335 
distribution,    effect    of    armature 

current  on,  135 
effect  of  neighboring  poles  on, 

126,  130 

effect  of  tooth  saturation  on,  132 
over  armature  surface,  124,  129, 

213,  221,  342 
leakage  (see  Leakage). 
Form  factor,  294 
Frequency  of  alternators,  238 

of  D.C.  dynamos,  78 
Friction   coefficient    (between   met- 
als), 52 

of  brushes,  182 
losses  (alternators),  317 
(brushes),  181,  227,  318 
(dynamos),  196 


Gauss  (definition),  13 
Generator  (see  Alternator;  dynamo). 
Gilbert,  11;  definition,  12 
GRAY,  PROF.  A.,  referred  to,  309 


HAWKINS  AND  WALLIS,  referred  to,  99 
HAY,  DR.  A.,  referred  to,  116,  132 
Heating     coefficient     (see     Cooling 

coefficient). 

intermittent,  of  magnet  coils,  46 
of     armatures,     coils,     etc.     (see' 

Temperature  rise). 
HELE-SHAW,  H.  S.,  referred  to,  116 
HOBART  AND  PUNGA,  referred  to,  307 
Homopolar      machines      (reference 

only),  83 

Horsepower   transmitted    by    mag- 
netic clutch,  52 

Horseshoe  lifting  magnet,  design   of 
53 


362 


INDEX 


Hysteresis  losses  in  armature  stamp- 
ings, 102 
teeth,  196 


Imagination,  value  of,  to  the  de- 
signer, 7 

Inclined  surfaces,  effect  of,  on  mag- 
netic pull,  30 

Inductance  of  A.C.  armature  wind- 
ings, 263,  288 

of  armature  end  connections,  304 
of  slot  windings  (turbo-alternator), 
329 

Inductor  (definition),  72  (see  also 
Armature  conductors). 

Inherent  regulation  (definition),  252 
(see  also  Regulation). 

Insulating  materials,  37,  95 

Insulation,  breakdown  voltages,  40 
of  armatures,  91,  96,  207,  258,  329 
maximum  allowable  temperatures 

for,  108 
thickness  of,  on  magnet  spools,  41 

Intensity  of  magnetic  field  (defini- 
tion), 12 

Intermittent  heating  of  magnet 
coils,  46 

Interpole  design  (numerical  exam- 
ple), 171 

Interpoles,   165,  advantages  of,  169 
calculation   of    ampere-turns    re- 
quired on,  174 

K 

KAPP,  DR.  G.,  referred  to,  110 
L 

LAMME,  B.  G.,  referred  to,  141 
Language,  importance  of  thorough 

study  of,  5 

Lap,  or  multiple,  armature  wind- 
ing, 86 

Leakage  factor  (definition),  28 
in  alternators,  300 
in  dynamos,  187 


Leakage  of  commutating  poles,  167, 

174 
Leakage  flux,  21 

effect  of,  in  saturation  under- 
load, 305 
in  armature  slots,  149,  281 

calculation  of,  160,  284 
in  circular  lifting  magnet,  67 
in  horseshoe  magnet,  57 
in  multipolar  dynamo,  187 
in  similar  designs,  27 
in  turbo-alternators,  349 
paths,  permeance  of,  in  air,  22 
LEHMANN,  DR.,  referred  to,  124 
Lifting    magnets    (see    Electromag- 
nets). 

LISTER,  G.  A.,  referred  to,  46 
Losses  at  brush  contact  surface,  181, 

182,  227,  318 
in  armature  conductors,  318 

stampings,  101,  104,  266,  331, 

356 

teeth,  102,  196,  211,  233,  318, 
331 

M 

McCoRMiCK,  B.  T.,  referred  to,  307 
Magnetic  circuit,    10;  fundamental 

equation,  14 
of  alternator,  299 
of  dynamo,  185,  228 
circuits  in  parallel,  18 
clutch,  design  of,  50 
flux,  11;  definition,  13 
force  (H),  definition  of,  12 
leakage  (see  Leakage  flux), 
pull  (formulas),  29 
Magnetization  curves  for  iron,   16, 

17,  18,  215 

Magnetizing  force  (H),  12 
Magnetomotive  force  (m.m.f.),   11; 

definition,  12 

distribution  of,  over  armature 
periphery,  128,  134,  342,  344 
Magnets  (see  Electromagnets). 
Magnet  windings,  calculation  of,  41, 

61,  66 

Materials    used    in    magnets    and 
dynamos,  32 


INDEX 


363 


Maxwell,  11;  definition,  13 
MENGES,  C.  L.  R.  E.,  referred  to,  155 
Mesh  connection,  244 
Mica,  commutator,  thickness  of,  94, 

177 
MOORE,   PROP.  C.  R.,  referred  to, 

101,  132 
MORDEY  flat  coil  alternator  referred 

to,  239 

MORTENSEN,  S.  H.,  referred  to,  307 
Moss,  E.  W.,  referred  to,  84 
Motors,  continuous  current,  70,  235 
MOULD,  J.,  referred  to,  84 
Multiple,  or  lap,  winding,  86 
Multiplex  armature  windings,  86 


Neutral  field,  or  zone  (definition), 

146,  283 
NOEGQBRATH,  J.  E.,  referred  to,  84 

O 

Oersted  (definition),  13 
Open-circuit     characteristic     curve 

(see  Saturation  curve). 
Output  formula  for  D.C.  dynamos, 
73,76 

of  three-phase  generator,  247 


PARSE  ALL  AND  HOB  ART,  referred  to, 

101 

PELTON  water-wheel,  referred  to,  241 
Peripheral  loading  (see  Specific  load- 
ing). 

velocity  of  alternator  rotors,  238 
of  commutators,  184 
of  dynamo  armatures,  77 
Permeability,  11,  15;  definition,  13 
Permeance,  11;  definition,  13 
between  concentric  cylinders,  26 
flat  surfaces,  24 
parallel  cylinders,  27 
curve,  123,  130 
of  air  gap    (slotted   armatures), 

116,  272 

of  leakage  paths  (formulas),  23 
Phases,  number  of,  239 


Plunger  magnet,  49 

Polar  coordinates,  use  of,  for  plot- 
ting irregular  waves,  294,  347 

Pole  arc,  74,  76,  204,  249 

Pole-face  windings,    170,   279,   322, 
336 

Pole,  length  of,  186,  227 

Pole   pitch    (definition),   74;    usual 

values,  80 

factors   determining  width   of, 
78,  248 

Pole  shoes,  188,  269 

Poles,  number  of,  80,  204,  241 

POWELL,  P.  H.,  referred  to,  116 

Pressure    rise    when    switching    off 
magnet  coil,  33 

Pressures,     usual,     in      commercial 
designs,  33,  248 

Properties  of  materials,  32 

Pull  exerted  by  magnets,  29,  50 


Quantity     of     air     required     with 

forced  ventilation,  112 
of  electricity,  70 

R 

Reactance,  armature,  304 

coils,  use  of,  355 
Reactive  voltage  drop  (slots),  288; 

(ends),  266,  304,  332 
Regulation,  effect  of  flux  distribu- 
tion on,  311 

inherent  (definition),  252 
of  alternators,  252,  301,  312,  352 
on  any  power  factor,  309 
on  zero  power  factor,  305 
Reluctance,  magnetic,  1 1 ;  definition, 

13 

Resistance  of  armature  windings,  97 
of  copper  wires,  36 
variation  of,  with  temperature,  36 
Rheostat,  shunt  field,  191 
Rise  of  pressure  on  switching-off  coil, 

33 
of  temperature  (see  Temperature 

rise). 

Rotor  of  turbo-alternators,  270,  272, 
335 


364 


INDEX 


Saturation  curve  for  alternator,  302, 

306,  310,  340,  355 
for  dynamo,  189,  190,  228 
Self-induction,    not    different   from 

other  induction,  155 
Series  field  winding,  190,  192,  231 

or  wave,  winding,  86 
Short-circuit  current  of  alternators, 

252,  308,  354 
Short  pitch  armature  windings,  85, 

160 

Shunt  winding — calculation   (dyna- 
mos), 191,  231 
with  two  sizes  of  wire,  42 
Simplex  armature  windings,  86 
SIMPSON'S  rule,  122,  217 
Single-phase  alternators,  280,  321 
Size  of  wire,  calculation  of,  in  mag- 
net coils,  41 

Sketches,  importance  of  neat,  4 
"  Skull  cracker,"  64 
Slot,  dimensions  of,    in   alternator, 

259 

in  dynamo,  92,  93,  207 
insulation  (see  Insulation  of  arma-  • 

tures). 
leakage  flux,  calculation  of,   160, 

284 
effect    of,    on    commutation, 

149 

"equivalent,"  161,  164,  285 
in  alternators,  281 
pitch  (definition),  92 
Slots,  number  of,  93,  260 
Space      factor       (definition),       39; 

(values),  40 
Sparking  at  brushes,  prevention  of, 

175,  225 

Specific  heat,  47 
Specific  loading  (definition),  74 

values  for  alternators,  250,  325 

for  dynamos,  76 

Speeds,  usual  (dynamos),  81;  (alter- 
nators), 241 
Star  connection,  244 
STEINMETZ,  DR.  C.  P.,  referred  to, 
141 


S.W.G.  wire  table,  35 
Symbols,  list  of,  xi 

use  of,  2 
Synchronous    A.C.    generators    (see 

Alternators). 


Teeth,  dimensions  of  (dynamos),  92, 

208;  (alternators),  259 
flux  density  in,  104,  119,  122,  197, 

209 
losses  in,  102,  196,  211,  233,  318, 

331 

number  of,  93,  260 
strength  of  rotor,  336 
taper,  losses  in,  196 

m.m.f.  calculations  for,  122 
Temperature,    internal,    of   magnet 

coils,  45 

rise  of  alternator  field  coils,  300 
of    armatures,    107,    112,    212, 

267,  357 

of  circular  lifting  magnets,  67 
of  commutator,  181,  227 
of  dynamo  field  coils,  193,  232 
of  iron-clad  coils,  46 
of  magnet  coils,  44,  47,  56,  62 
of     totally     enclosed     motors, 

236 
Third  harmonic,  effect  of  Y  and  A 

connections  on,  246 
THOMPSON-RYAN,  referred  to,  170 
Three-phase  generators   (see  Alter- 
nator). 
Time,  value  of,  in  engineering  work, 

64 
Tooth  saturation  (comparison  D.C. 

and  A.C.),  271 

Tractive  force  (magnetic  pull),  29 
of  cylindrical  short-stroke  mag- 
nets, 50 

Tube  of  induction,  14 
Turbo-alternator,  design  sheets  for 

8,000  k.v.a.,  322 

Turbo-alternators,   output  of  mod- 
ern, 320 

(rotor  construction),  270 
(usual  speeds),  242,  320 


INDEX 


365 


U 

Undercutting  commutator  mica,  181 
UNDERBILL,  C.  R.,  referred  to,  48 
Units,  conversion  of  centimeter  to 
inch,  2 

V 

Vector  diagrams  of  alternator: 
For  determining  inherent  regula- 
tion, 315 

Of  short-circuited  armature,  308 
On  lagging  power  factor,  289,  290, 

311,  334 

On  zero  power  factor,  287,  307 
Velocity  of  air  in  cooling  ducts,  268, 

357 

peripheral     (see     Peripheral     ve- 
locity). 
Ventilating   fans,    power   necessary 

to  drive,  107,  318 
Ventilation  of  armatures,  105,  261, 

351 
Virtual  values  of  irregular  waves, 

294,  346 

Voltage  (see  also  E.m.f.). 
drop  at  brush  contact,  179 
generated  by  cutting  flux,  71 
rise  when   switching  current  off 
coil,  33 


Voltage,usual,  of  A.C.  generators,  248 
Voltages  to  be  considered  in  connec- 
tion with  insulation  problems,  33 


W 


WADDELL,  DR.  J.  A.  L.,  referred  to,  5 
WALKER,    PROF.    M.,    referred    to, 

107,  255 

Wave  form  (see  E.m.f.  wave  form). 
Wave  winding  (dynamos),  86 
Wedge  for  rotor  slots,  336 
Whirling  speed  of  rotors,  326 
Windage,  loss  due  to,  196,  317,  356 
Winding,  depth  of,  to  give  required 

excitation,  43 
on  magnet  limbs,  55 
shunt  coils  with  two  sizes  of  wire, 

43 

Windings  (see  Armature  field,  mag- 
net, windings). 
Wire  (see  also  Copper  wire), 
enamelled,  38 
tables,  34,  35 


Y  connection,  244 

Yoke  ring,  dimensions  of,  228 


JUL  7    1933 
30  1953 


r/r 


338041 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


